Richard M. Smith, the president of the Pinkerton Foundation, which gave the fund $1.4 million from 2012 to 2014 for a three-year pilot of Summer Quest, said the government could take risks with private funds that it could not with tax dollars. He added that private money helped get programs up and running quickly, because the rules of public procurement did not apply.

“The great value of the role of private philanthropy is that it can experiment with potential solutions to major problems that would be politically too controversial if attempted with public money,” he said.

Mr. Klein also used the private money he and Ms. Kennedy raised, some of which came through the fund and some through other nonprofits, to finance the opening of new small high schools, to create a training institute for new principals and to undertake other experiments in overhauling the school system, some of which, including the small high schools, the current administration has not embraced.

Image The schools chancellor, Carmen Fariña, like Mr. de Blasio, is not as connected to a world of wealthy donors in the same way that her predecessor was. Credit Karsten Moran for The New York Times

Denis Calabrese, the president of the Laura and John Arnold Foundation, which gave the fund $1.25 million during the Bloomberg years to finance a pilot teacher evaluation system, said philanthropists who supported charter schools had probably been turned off by Mr. de Blasio’s fight last year with Eva S. Moskowitz, the founder of Success Academy, the city’s biggest charter school network, over space for three of its schools. (The city ultimately gave the schools space.)

“That was the most visible thing that funders saw in terms of gauging the approach,” Mr. Calabrese said.

Peter Sloane, the chairman and chief executive of the Heckscher Foundation for Children, which gave the fund close to $1.2 million from 2006 to 2008 for a free breakfast and lunch program during the summer, said that if the fund were having difficulty raising money, “it may reflect a more basic discomfort with the education agenda of this administration.”

About half the money counted as raised this year comes from two large, multiyear grants made during the 2012 fiscal year, from the Open Society Foundations and the Wallace Foundation, the first to support programs aimed at improving outcomes for young black and Latino men, and the second for principal development.