On Martin Luther King’s Birthday in January 2017, Donald J. Trump, then the president-elect, welcomed a group of civil rights leaders, led by Dr. King’s eldest son, into his office in Trump Tower.After a tour of Mr. Trump’s celebrity curio collection (Shaquille O’Neal’s sneakers, size 22, were a highlight), the visitors presented him with a proposal intended to prevent state voter identification laws from disenfranchising people of color.The delegation had low expectations. Mr. Trump had championed the lie that President Barack Obama was not born in America and, in their view, played to racial fears during the 2016 campaign. He quickly dashed even those modest hopes. Low turnout among Black voters, Mr. Trump declared, had helped him defeat Hillary Clinton.“Many people didn’t go out — many Blacks didn’t go out — to vote for Hillary because they liked me. That was almost as good as getting their vote,” Mr. Trump said, lowering his voice to say the word “Blacks,” on a recording provided by a meeting participant and confirmed as authentic by three others. (A White House spokesman did not dispute the veracity of the recording.)Mr. Trump promised he would seriously consider their proposal. It went nowhere.“I will be better to the African-American people than anybody else in this room,” he declared just before heading down the elevator to appear before the cameras with his guests, according to the recording, which was shared with several news organizations last month.To Mr. Trump, this was little more than a photo op: Two former aides recalled that he wanted to be seen with a group of Black leaders to rebut an assertion made by Representative John Lewis, the late civil rights paragon, who at the time had said he did not “see this president-elect as a legitimate president.”As the 2020 campaign hits the homestretch, Mr. Trump has been claiming that he is the best president for Black Americans since Abraham Lincoln and papering over his history of racist remarks by having Black supporters at the Republican convention back his boast that he “is the least racist person in the world.”In fact, Mr. Trump has hired very few Black officials to positions of authority in the White House and for his re-election effort. And his campaign has stoked racial divisions to an extent not seen since George Wallace’s run in 1968. He has tried to block or hamper efforts to expand ballot access. He has said Black people were “too stupid” to vote for him, according to his estranged former attorney, Michael Cohen.When asked about Mr. Cohen’s charge, a White House spokesman emailed a statement by Press Secretary Kayleigh McEnany calling Mr. Cohen, “a disgraced felon and disbarred lawyer, who lied to Congress.”And as the 2017 meeting illustrates, many of Mr. Trump’s interactions with Black leaders have followed a similar pattern: He has turned opportunities for reconciliation, or even to debate policy differences, into empty-calorie encounters in front of the cameras, according to interviews with more than 30 Black officials, civil rights leaders, and former and current administration officials.Trump campaign officials told reporters last week that they were working hard to slightly exceed their performance with Black voters in 2016 when they won about 8 percent of their vote. The president has also tried to mollify white moderates who might be turned off by his racial rhetoric.The result is a jarring, split-screen approach: The president talks up his friendships with Black Americans (often famous ones like Kanye West) while running a campaign whose objective is to frighten white suburban voters into thinking a Biden presidency will bring problems from inner-city America to their front lawns.“The goal of the these things was just to get him through the meeting, then get the picture,” said Anthony Scaramucci, the former Trump aide, now an outspoken critic of the president, who set up several meetings between Mr. Trump and Black sports figures, including the football legend Jim Brown.At first, some Black leaders seemed open to the president’s entreaties. But they said any chance of meaningful communication with Mr. Trump ended, with the finality of a slammed door, when he declared there were “very fine people on both sides” after white supremacists rioted in Charlottesville, Va., in August 2017.For Martin Luther King III, Dr. King’s son, the critical moment came a few minutes into the January 2017 session, when Mr. Trump falsely claimed that “hundreds of thousands of people” had crossed the border to illegally cast ballots for Mrs. Clinton.“My hope, because I believe in the human spirit in most people, is that he would recognize there was an adjustment that had to be made,” Mr. King said in an interview. “But he has no interest in doing anything for anybody who does not support him. The vast majority of Black people do not support him, so he has nothing for the majority of Black folks.”Theodore Mukamal, a civil rights attorney who recorded the meeting, said he had gone in assuming that the president wanted to move beyond the rancor of the election.“It was a disaster,” said Mr. Mukamal. “There was no follow through, and it turned out the opposite of what we could have possibly predicted.”During an interview with Mr. Trump for a new book, the journalist Bob Woodward asked the president if he felt a need to “understand the anger and pain” of Black Americans.“No,” Trump replied. “You really drank the Kool-Aid, didn’t you? Just listen to you. Wow. No, I don’t feel that at all.”Ja’Ron Smith, a deputy assistant to the president, said that Mr. Trump’s efforts have been geared at ensuring “hardworking patriots of the Black community are forgotten no longer,” rather than winning over the critics.“For decades Democratic leaders have stood behind empty promises when it came to Black Americans,” wrote Mr. Smith in an email. “The polices of stagnation and governmental dependence have been dismantled and replaced with economic prosperity and opportunity.”Mr. Trump continued to reach out after he was elected. In March 2017, Omarosa Manigault Newman, the former “Apprentice” star-turned-White House adviser, invited members of the Congressional Black Caucus to the White House to discuss their policy agenda.Mr. Trump ushered them, cheerfully, into the Cabinet Room. They sat across the table, grimly, with blue binders. Representative James Clyburn, a South Carolina Democrat, recalled that Mr. Trump interrupted three times to suggest they relocate to the Oval Office.Finally, as Mr. Clyburn began his pitch for a national infrastructure and broadband plan, Vice President Mike Pence popped in to announce that they needed to vacate the room to make way for another meeting, Mr. Clyburn said.“Let’s go!” Mr. Trump said.The group followed him into the Oval Office, but congregated near the fireplace; Representative Cedric Richmond, a Louisiana Democrat who chaired the caucus, turned down Mr. Trump’s request for a photo but he agreed to photographs showing the caucus negotiating — far from Mr. Trump’s favorite snapshot destination, the Resolute Desk.“I know his type,” Mr. Clyburn said in an interview. “He has been as caustic as he can be. Trump insults in public and then makes a private phone call to you to say that is really not what he meant. I have been around those kind of white people all my life.”Mr. Trump continued to pursue high-profile meetings. At the same time, his political appointees had begun to systematically scuttle enforcement of civil rights laws in education, housing, health care and at the Justice Department.The president seemed to place particular emphasis on systematically erasing the legacy of the first Black president; one of his aides printed out a checklist of Mr. Obama’s accomplishments to attack them, one by one.Mr. Richmond declined an invitation for another meeting later in 2017, writing that doing so would be pointless, “given the actions taken by your administration since our first meeting.”Mr. Trump’s supporters believe that the failure of the Democratic Party to fulfill its promises to Black Americans has created an opening for the president’s unorthodox approach.Wayne Dupree, a conservative talk radio host in Baltimore, said that “things wouldn’t have been any different” if Mr. Trump had made a more concerted effort at outreach.Darrell Scott, a Cleveland-area pastor who co-founded the Trump campaign’s diversity panel, said the lack of a “constructive relationship” is not Mr. Trump’s fault.“People always ask me why there are no Blacks in the White House,” he said. “This is the spoils system, if someone comes on board in the campaign, they go to the White House. But the Blacks who came onto the campaign were labeled ‘coons,’ ‘Uncle Toms’ and ‘sellouts,’ so Blacks were not applying for positions.”The most important back channel for communication has been Mr. Smith, who has played a key role in three achievements most frequently touted by the president: a tax credit program aimed at increasing investment in low-income neighborhoods, increased funding for historically Black colleges and universities and the bipartisan criminal justice reform bill.Mr. Smith — who resisted efforts to play a bigger public role before agreeing to speak at the convention last month — has also quietly reached out to some civil rights groups even as the president was hurling gasoline-on-the-fire tweets, according to a White House official with knowledge of his actions. He is also one of the few White House officials pushing for a response to address disparities in health outcomes for people of color during the pandemic, and tried, with limited success, to broker bipartisan police reform legislation, another official said.The president sought to project a more empathetic image in the immediate aftermath of George Floyd’s killing in Minneapolis in June. Mr. Smith’s efforts notwithstanding, Mr. Trump has not substantively addressed police violence. In June, he convened a group of Black conservatives and talk-radio hosts to the White House for a discussion in front of the cameras. Mr. Trump began with a rosy assessment of race relations and economic opportunity.Sonnie Johnson, a conservative radio host, challenged him. “The Black community is not OK — I understand the perspective and the desire to put out this talking point, but it’s not,” she said.When she was done, Mr. Trump declared, “Good job.”He did not address her specific concerns.The two Black people invited to a televised round table with Mr. Trump in Kenosha, Wis., held in the wake of the police shooting of Jacob Blake, were James E. Ward Jr. and Sharon Ward, husband-and-wife pastors from a church in Skokie, Ill., who are close to Mr. Blake’s family.When a reporter asked them if police violence was a systemic issue, Mr. Trump, perhaps thinking the question was for him, cut them off to say, “I think the police do an incredible job.”Annie Karni contributed reporting.