During the administration of Mr. Bloomberg, a political independent, the fund financed a wide range of activities, like improvements to school libraries and performance spaces and bringing technology into classrooms. The fund was also used to try out new programs. Some pilot initiatives financed by the fund, such as the program to distribute summer meals, were later continued or expanded with public money. Others, including an experiment with giving teachers performance bonuses if their schools met certain academic targets, were deemed unsuccessful and discontinued.

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.