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.

As the budget process progressed, updated revenue projections showed that many of the cuts were unnecessary. Both fiscally conservative and liberal good government groups and the Independent Budget Office called City Hall’s revenue projections unnecessarily pessimistic and inaccurate. But the mayor ordered agencies to slash their budgets anyway.

Nathan Gusdorf, the director of the left-leaning Fiscal Policy Institute, said the mayor’s “unduly pessimistic revenue forecasts” were “fiscally irresponsible” and had resulted in hiring freezes and the elimination of jobs that helped the city run smoothly.

“As the cost of living rises and our city loses working and middle-class families, the mayor should prioritize deeper investments in child care and affordable housing to keep New Yorkers here,” Mr. Gusdorf said, “rather than insisting on budget cuts that will only drive more families out.”

Justin Brannan, the chairman of the Council’s Finance Committee, said he and his colleagues never doubted that the city had enough revenue to restore most of the mayor’s budget cuts, citing the need to invest in housing, early childhood education, arts and culture and mental health.

“If we want to make sure New York City remains the capital of the world,” Mr. Brannan said, “we’ve got to keep investing in it.”

The budget also includes roughly $2 billion for affordable housing and restores funding for arts programs and half-price MetroCards for poor New Yorkers.

A package of roughly $100 million was included for early childhood education and programs to improve the system so it can more equitably serve all New Yorkers, including children with disabilities.

About $20 million will pay for additional preschool seats for 3-year-olds, which is known as 3-K. Other funding will fill vacant seats and clear wait lists for children who receive special education services, and a biweekly working group will focus on addressing the problems.

Some 3-K supporters were disappointed that the program did not receive enough funding to make it truly universal.

“Parents are grateful to the New York City Council for their Herculean efforts in achieving a budget that rolls back some of the mayor’s cuts to 3-K,” said Rebecca Bailin, executive director of New Yorkers United for Child Care. “Despite these steps, families are still facing millions in unnecessary cuts to 3-K.”

Some Democrats who are considering running against Mr. Adams next year have criticized his management of the city, arguing that his budget cuts have sowed confusion and hurt working-class New Yorkers. The mayor’s cuts to early-childhood education programs, for example, are expected to be a major issue in the upcoming primary.

“The mayor should be laser-focused on making our city more livable and more affordable,” said Zellnor Myrie, a state senator from Brooklyn who is exploring a mayoral run. “Instead, his mismanagement and budget cuts are making it harder for families in every way.”

Scott Stringer, the former city comptroller who is exploring a primary challenge against Mr. Adams, said the mayor’s questionable revenue projections made it feel like the city had regressed to the “bad old budget days of the 1970s when the city was on the edge of bankruptcy” and lamented that the budget had harmed families and children in particular.

Jessica Ramos, a state senator from Queens who is also considering running for mayor, called the budget “mediocre and uninspired” and said the budget process should “evolve past public gaslighting.”