Gov. Greg Abbott of Texas on Wednesday kicked off a high-stakes political fight over the state’s future, formally announcing a special session of the Legislature in which he and fellow Republicans will try to push Texas further to the right on issues like elections and voting, transgender rights and how racism is taught in schools.The special session, set to begin on Thursday, follows an already ultraconservative legislative session this spring, when the Republican-dominated Legislature passed a near-ban on abortion and a law permitting the carrying of handguns without permits, running roughshod over protests from Democrats, business coalitions and civil rights groups in an often strictly party-line manner.But the Legislature failed to pass one of the governor’s signature priorities for the session — a sweeping election overhaul bill that would have been one of the most restrictive voting laws in the country — when Democratic state lawmakers staged a dramatic late-night walkout that deprived the House of a quorum and temporarily killed the bill.Republicans’ new election overhaul bill in Texas, a state which already has some of the nation’s strictest voting rules, will be the first to come before a state legislature since the Supreme Court’s ruling last week to uphold two voting restrictions in Arizona. That decision significantly elevated the threshold for whether a voting measure constitutes a violation of Section 2 of the Voting Rights Act, which prohibits discriminatory voting practices.While the second attempt to pass voting measures will be perhaps the most closely watched legislative battle when the session convenes on Thursday, Mr. Abbott also called for the Legislature to take up measures combating perceived “censorship” on social media platforms; banning the teaching of “critical race theory” in public schools; further limiting abortions; putting in place new border security policies; and restricting transgender athletes from competing in school sports.Mr. Abbott is also seeking more dedicated funding for property tax relief and cybersecurity.The governor is up for re-election next year, when he will face a challenge from his right in the Republican primary race. He has also been seen in Texas as laying the groundwork for a potential presidential bid in 2024.“The 87th Legislative Session was a monumental success for the people of Texas, but we have unfinished business to ensure that Texas remains the most exceptional state in America,” Mr. Abbott said in a statement. “Two of my emergency items, along with other important legislation, did not make it to my desk during the regular session, and we have a responsibility to finish the job on behalf of all Texans.”Lt. Gov. Dan Patrick, who also serves as president of the State Senate, pledged to complete Mr. Abbott’s agenda, but he made clear that a voting bill would be his, and therefore the Senate’s, top priority.“We are ready to pass all of the legislation on @GovAbbott’s Special Session call starting with #SB1 — Election Security,” Mr. Patrick wrote on Twitter. “Hearings will begin Saturday.”The agenda is sure to inflame Democratic state lawmakers who have already been angered by the Legislature’s rightward turn this year. And even some Republicans remain miffed at the governor’s decision last month to veto funding for the Legislature as a punishment for the Democrats’ decision to flee the Capitol over the voting bill.While lawmakers, who receive only a token stipend for their role, are not particularly affected by Mr. Abbott’s move, many staff members whose salaries are dependent on that money remain in limbo, with funding set to expire in September. The issue of funding the Legislature is also on the agenda for the special session.“What we think he’s done is definitely unconstitutional,” said State Representative Rafael Anchía, a Democrat from the Dallas area. “He’s trying to blackmail us to pass his agenda by defunding the legislative branch.”State Representative Chris Turner, the chair of the House Democratic caucus in Texas, accused the governor of leveraging the legislative session for his own political gain.“The governor’s agenda for the special session shows he is more concerned with pandering to die-hard Trump supporters and right-wing extremists than he is with serving everyday Texans,” Mr. Turner said in a statement.For Democrats, the inclusion of more funding for a border wall was proof that Mr. Abbott was playing strictly to the Republican base, because he had already announced that he would set aside $250 million from the state’s general revenue as a down payment for a border wall and asked online donors to foot the rest of the bill. A program manager would then determine the eventual size and total cost of the project.“A lot of people in Texas, they seem to think that there really isn’t a real border at the U.S.-Mexico border, that people are just walking over,” said State Senator José Menéndez, a San Antonio Democrat. While crossings have increased to levels not seen in years, Mr. Menéndez said, a wall would have little consequence as long as people continue to leave their homes in Central America to escape poverty, crime and corruption. “He’s trying to win that Trump base.”And that effort, Democrats said, came at the expense of other, more pressing issues in Texas.“I’m furious and I’m embarrassed for the state,” said State Representative Erin Zwiener, who argued that the governor was again trying to bar transgender student athletes from competing in sports while neglecting what she saw as needed reforms to health insurance or regulations to prevent another power grid failure, like the one that killed more than 100 Texans in February.“None are those are the issues we are focusing in on,” she said. “Instead we’re talking about a bill that attacks some of our most vulnerable children to make a culture war point.”The breadth of the agenda took even some Republicans by surprise.“My first take is, we’re going to be busy,” said State Senator Jane Nelson, one of the chamber’s senior members and the chair of the Senate Finance Committee. “This is a much longer call than I think most of us expected.”Ms. Nelson, a North Texas Republican who announced this week that she would not seek re-election after 28 years of service, said she was guardedly hopeful that lawmakers might be able to avoid an explosive confrontation over the voting bill.“What I am hoping is now that everybody’s had some time to get some sleep and think through what the controversial points were,” she said, “I think we’ll be able to work something out. I really do.”The Battle Over Voting RightsAfter former President Donald J. Trump returned in recent months to making false claims that the 2020 election was stolen from him, Republican lawmakers in many states have marched ahead to pass laws making it harder to vote and change how elections are run, frustrating Democrats and even some election officials in their own party.A Key Topic: The rules and procedures of elections have become central issues in American politics. As of May 14, lawmakers had passed 22 new laws in 14 states to make the process of voting more difficult, according to the Brennan Center for Justice, a research institute.The Basic Measures: The restrictions vary by state but can include limiting the use of ballot drop boxes, adding identification requirements for voters requesting absentee ballots, and doing away with local laws that allow automatic registration for absentee voting.More Extreme Measures: Some measures go beyond altering how one votes, including tweaking Electoral College and judicial election rules, clamping down on citizen-led ballot initiatives, and outlawing private donations that provide resources for administering elections.Pushback: This Republican effort has led Democrats in Congress to find a way to pass federal voting laws. A sweeping voting rights bill passed the House in March, but faces difficult obstacles in the Senate, including from Joe Manchin III, Democrat of West Virginia. Republicans have remained united against the proposal and even if the bill became law, it would most likely face steep legal challenges.Florida: Measures here include limiting the use of drop boxes, adding more identification requirements for absentee ballots, requiring voters to request an absentee ballot for each election, limiting who could collect and drop off ballots, and further empowering partisan observers during the ballot-counting process.Texas: Texas Democrats successfully blocked the state’s expansive voting bill, known as S.B. 7, in a late-night walkout and are starting a major statewide registration program focused on racially diverse communities. But Republicans in the state have pledged to return in a special session and pass a similar voting bill. S.B. 7 included new restrictions on absentee voting; granted broad new autonomy and authority to partisan poll watchers; escalated punishments for mistakes or offenses by election officials; and banned both drive-through voting and 24-hour voting.Other States: Arizona’s Republican-controlled Legislature passed a bill that would limit the distribution of mail ballots. The bill, which includes removing voters from the state’s Permanent Early Voting List if they do not cast a ballot at least once every two years, may be only the first in a series of voting restrictions to be enacted there. Georgia Republicans in March enacted far-reaching new voting laws that limit ballot drop-boxes and make the distribution of water within certain boundaries of a polling station a misdemeanor. And Iowa has imposed new limits, including reducing the period for early voting and in-person voting hours on Election Day.Key members of the Republican leadership in the Legislature, including Dade Phelan, the speaker of the House, and State Senator Bryan Hughes, the author of his chamber’s version of the voting bill last session, did not respond to requests for comment.Some Democrats also feared that the inclusion of measures on hot-button culture war issues, after similar bills had already died in the Legislature, could serve as a distraction from their effort to protect voting rights.“Make no mistake, one of the ways to minimize the import of voting rights is to surround it with hyperpartisan red meat,” said State Representative Trey Martinez Fischer, a Democrat who accused Mr. Abbott of using the legislative session to help his primary re-election campaign.The legislative acrimony came to head in late May over Republicans’ omnibus voting bill, which, after months of debate, was finished behind closed doors by G.O.P. legislators and lawyers in a conference committee while Democrats were kept in the dark.The final bill included a raft of restrictions on voting and elections, including new limits on absentee balloting, broad new autonomy and authority for partisan poll watchers, and stricter punishments for election officials who are found to have made errors. The bill also would have banned drive-through voting and 24-hour voting, which were used for the first time during the 2020 election in Harris County, home to Houston and many Democratic voters.Two late additions to the bill — a shortening of voting hours on Sundays that seemed intended to limit the popular “Souls to the Polls” programs of Black churches, and a provision that would make overturning elections easier — particularly enraged Democrats.Both Democrats and Republicans expect the initial version of the voting bill in the special session to be similar to the one that failed in May, though they also expect the provision on overturning elections to be removed. Even some Republican lawmakers had expressed concerns that the provision was included in the earlier final bill.David R. Montgomery contributed reporting from Austin, Texas.