Reopen N.Y.C. Libraries on Sundays? Yes. Free 3-K for All? Not Quite.

After months of tense and protracted negotiations, Mayor Eric Adams and City Council leaders announced on Friday that they had reached agreement on a $112.4 billion budget for New York City that restored many of the mayor’s proposed cuts, including to libraries and cultural institutions.

But other key programs were not made whole, including a popular and free preschool program for 3-year-olds.

This budget is particularly significant for Mr. Adams, a Democrat who is running for re-election in a competitive primary next June. Mr. Adams has insisted that major budget cuts were necessary to help offset the costs of the migrant crisis, new union contracts for city workers and the ending of federal pandemic aid.

But Council leaders and a wide range of advocates have argued that the cuts would make life harder for New Yorkers at a moment when the city was increasingly unaffordable. Groups rallied on the steps of City Hall to call for funding for libraries and preschools and enlisted celebrities such as Hillary Clinton and Rachel Griffin Accurso, a children’s entertainer known as Ms. Rachel.

Library leaders said on Friday that $58 million in restored funding would allow them to reopen branches on Sundays and to remain open on Saturdays. They added that Sunday reopenings would begin at some branches “in the coming weeks,” returning to the same hours of operation before cuts forced the closures in November.

The fight over the libraries was emblematic of the deep divide between the mayor and representatives of the Council speaker, Adrienne Adams. The two sides could not agree on basic revenue estimates and offered vastly different visions for the city. Neither got everything they wanted.