“We are going up the ambition scale here,” he said of his next goals: reducing homelessness, improving mental health services and addressing the dirt and disorder that tarnish the quality of New Yorkers’ lives.

The de Blasio administration wants to combine those goals with an effort to overcome the mayor’s image as a leader more interested in making grand pronouncements than filling potholes. It was a perception shaped by numerous missteps in Mr. de Blasio’s first two years in office: late arrivals to events, public comments that alienated an already skeptical police force, a feud with Gov. Andrew M. Cuomo, his fellow Democrat.

That image, which Mr. de Blasio has worked hard to shed, lent credibility to the notion that he could be vulnerable in the 2017 election, with potential challenges from both the right and left. But before the investigations came to light, Mr. de Blasio enjoyed rising popularity in polls, with no prominent Democrat emerging as a serious rival.

Now the inquiries — as well as a race-tinged joke the mayor made during a skit he performed with Mrs. Clinton on Saturday — have put Mr. de Blasio back on his heels.

“He appears to be the type of political leader that those type of allegations accrue to,” said Don Levy, director of the Siena Research Institute, which conducts political polls. “A few years into his administration, his armor has taken a few hits and his accomplishments at this point just don’t seem to have turned the tide on a growing narrative of his not being effective.”

Mr. de Blasio has said he was not immediately informed about a separate federal investigation, also by the United States attorney for the Southern District, Preet Bharara, into the New York City Housing Authority, which began turning over millions of documents last year.

Likewise, the mayor said he first heard of the federal corruption investigation involving the donors, Jeremiah Reichberg and Jona S. Rechnitz, from news reports this month. The police commissioner, William J. Bratton, has said the federal authorities told him of the widening investigation in 2014. (At the time, people briefed on the matter have said, the focus was on Philip Banks III, then the highest-ranking uniformed officer, and Norman Seabrook, the head of the city correction officers’ union.)