The mayor’s defenders are quick to note that he has been more constrained politically than might be expected in a city so liberal. The most severe limit on his capital is the natural tension between city and state governments in New York — the state effectively controls several key elements of city life, including the moldering subways — and the unnatural tension that has colored de Blasio’s relationship with Gov. Andrew Cuomo. Despite coming from the same party, the two have maintained a lengthy feud premised as much on personality as on policy. In 2015, amid a series of often-petty spats, de Blasio accused the governor of carrying out a “vendetta.” The conflict has cooled some, which is itself a damning signal that Cuomo, the typically pitiless alpha, considers de Blasio effectively vanquished.

More damaging for de Blasio’s local clout, several of his recent decisions have run afoul of the party’s progressive wing, with which he has long identified. In a rare joint endeavor with Cuomo earlier this year, the mayor tried to court Amazon with a subsidy-rich package to plant its second headquarters in New York, before the company backed out amid fierce backlash from the left over the proposed tax breaks and negotiation process. Last year, de Blasio supported Joe Crowley, king of the Queens political machine, in his campaign to defend his House seat against Ocasio-Cortez. In June, de Blasio allowed rivals in the presidential race to get to his left in his own city: In a primary for Queens district attorney, Sanders and Elizabeth Warren endorsed Tiffany Cabán, a millennial Latina who campaigned as a hard-left “decarceral prosecutor.” De Blasio stayed neutral.

Even allies have often kept a distance. Last year, when the actress Cynthia Nixon, a friend of the mayor’s, mounted a long-shot progressive challenge to Cuomo, her campaign considered the wisdom of formally courting de Blasio’s support. But an internal survey found that this would not have helped her, and the idea was abandoned, according to people familiar with the race.

Charlie Rangel, the 89-year-old former 23-term congressman from Harlem, told me he had never encountered a politician as isolated as de Blasio, who managed his 1994 re-election. “I don’t know a goddamn person that’s his buddy — nobody,” Rangel said. “And everybody, even Giuliani, I know who his buddies are.” (Rangel added dryly that he had “completely forgotten” that the mayor declined to endorse him in his final House primary in 2014, after Rangel was censured for ethical violations and younger candidates started gunning for him.)

This isolation has at times extended to the mayor’s own workplace. De Blasio has alienated even some close aides with fits of highhandedness, bellowing driving instructions at his security detail or telling off senior staff, at times dismissing their guidance with slashing asides like “Nobody ever voted for you.” In a meeting at a hospital in the Bronx during an outbreak of Legionnaires’ disease in 2015, de Blasio flew into a rage at his buildings commissioner, Rick Chandler, over the pace of cooling-tower inspections, according to two people present. “You’re [expletive] bureaucrats,” the mayor shouted, rising to storm out. But the theatrical force of the move was undercut by a scheduling flaw: De Blasio still had to give a news conference on-site. Officials found another conference room for him to sit in until it was time to start. (Goldstein, the mayor’s press secretary, attributed subsequent improvements in cooling-tower regulations to the mayor’s urgency.)

Other efforts to project authority have likewise faltered. As he took office, de Blasio instructed top officials to assemble each morning at 8 a.m. to discuss the business of the day. The mayor stopped coming within weeks, according to several attendees. Anecdotes like this have fueled persistent insinuations that de Blasio is lazy, a charge to which he is particularly sensitive. Some friends consider this reputation to be almost entirely a function of race. Peter Ragone, a longtime adviser, cites internal surveys showing that black voters reject media portraits of the mayor, while white voters often believe the worst about him, at times coming under the false impression that crime has risen during his administration. “If you are a person on the Upper East Side or the Upper West Side,” Ragone told me — “or The Times’s newsroom,” he added pointedly — “your perception of de Blasio is a mirror image of what people of color think. You know what people of color love? Jobs and not getting frisked by the cops.” And as a crusader for these constituents and a member of a biracial family, the argument goes, de Blasio has absorbed the racist stereotypes typically applied to black politicians.

J. Phillip Thompson III, a deputy mayor under de Blasio who wrote a book about black mayors (and who is black himself), told me de Blasio’s political challenges ring familiar, comparing past accounts of Dinkins’s penchant for tennis and high fashion to media skepticism about de Blasio’s interest in the job. Thompson suggested each had been manufactured to depict the men as lightweights. (When I put this theory to Dinkins, New York’s only black mayor, he sounded unconvinced. “I don’t think so,” he told me. “If you hold office, you’re going to have some criticisms of that sort.”) I told Thompson I had spoken to many close associates of de Blasio’s who did question his affection for the role. Goldstein, who was monitoring the interview, cut in.