Mr. de Blasio’s administration has said little about the closed hospital since he took office, other than calling a news conference in early 2014 to celebrate a deal that would have required proposals for the LICH site to include “maximum health care.”

But his aides have long given up on a full-service hospital at the site and are instead trying to nail down the package of affordable units. “The whole situation became a total mess,” said one top adviser to the mayor.

[. . .]

But as Mr. de Blasio, a liberal Democrat, approaches the midpoint of his own term as mayor of New York City, he has executed an abrupt and deliberate shift in posture.

Over the last month, Mr. de Blasio has repeatedly hailed Mr. Bloomberg, a political independent, as a champion for the city, doing so in unsolicited, sometimes unexpected, ways. He applauded Mr. Bloomberg for his “vision and for the energy” spent in expanding the No. 7 subway line; he credited him for a hard-fought victory in winning mayoral control of the public schools; and he even took to Twitter to defend him against the National Rifle Association.

The timing of Mr. de Blasio’s overtures is no accident: The mayor and his advisers have increasingly come to see the rift between Mr. de Blasio and Mr. Bloomberg as a damaging and unnecessary political liability — especially at a time when Mr. de Blasio’s clashes with Gov. Andrew M. Cuomo, a fellow Democrat, have hurt his ability to push through important pieces of his agenda.

[. . .]

City Hall officials say that with a presidential election looming, Mr. de Blasio cares deeply about focusing the political dialogue on policies that he believes can help combat inequality, like eliminating tax loopholes for the income of some investment managers.

While the mayor has received criticism for his time spent outside New York, much of his national travel, aides say, is ultimately rooted in aiding the city. On trips to Washington, for instance, Mr. de Blasio often meets with federal law enforcement officials, visits the White House, and has rallied to increase infrastructure funding for New York.

The collaboration between cities and advocacy groups, however, has not always gone smoothly, according to a review of hundreds of pages of public records, obtained through Freedom of Information requests in nine major cities, including Chicago, Los Angeles, Philadelphia and San Francisco.

In January, New York officials clashed sharply with an outside counsel over the group’s first major project: an amicus brief that Cities United filed in support of Mr. Obama, in federal court in Texas. At the recommendation of the National Immigration Law Center, the coalition retained a Los Angeles firm, Andrade Gonzalez, to prepare the document.

Sean Andrade, a founder of the firm, said he suggested that Cities United list mayors from Texas as the major litigants in the case. Officials in New York disagreed, preferring to name New York City and Los Angeles as the chief litigants.

To Mr. Andrade’s dismay, New York officials also demanded that his firm’s name be removed from the brief.

[. . .]

The spat, however, raised lingering doubts: In an email shortly afterward, Mr. Noorani, president of the National Immigration Forum, asked a top Los Angeles city official for her assessment of a Cities United proposal, “given the incredible amount of drama they kicked up on the amicus.”

But conditions in Times Square have recently come under scrutiny, with some New Yorkers complaining about the proliferation of street performers — including, most notably, topless women wearing body paint — who are said to be accosting pedestrians for tips for posing for photographs.

Mr. de Blasio has been keen to demonstrate that he is addressing the concerns, and on Thursday he announced a task force to consider ideas on how to better prevent activities that the city deems illegal or harmful to the area’s quality of life.

But the mayor, at a news conference in Queens, surprised many urban planners when he said he would give “a fresh look” to whether the pedestrian plazas should remain.