Mayor Zohran Mamdani announced on Monday that New York City will expand free child care for 2-year-olds to Staten Island and across the Bronx as the city moves to add 10,000 seats next school year.

Mr. Mamdani ran for mayor on the idea of providing universal child care as a way to help families afford to stay in the city. It is his signature policy and will be the most expensive and difficult to carry out among his broader affordability proposals.

The 2-K program started this school year with 2,000 seats in four largely working class areas of the city. A major expansion could test the managerial competence of Mr. Mamdani’s administration, which has struggled to address payment delays for child care providers.

Mr. Mamdani on Monday visited a trampoline park in the Bronx with Gov. Kathy Hochul to highlight the neighborhoods where the new seats will be offered. More than half of the city’s 32 school districts are represented in the expansion, which includes all of Staten Island and the Bronx.

“From Co-op City in the Bronx to Staten Island’s South Shore, we’re bringing 2-K to every borough and making New York City more affordable for working people,” Mr. Mamdani said in a statement.

Wealthier neighborhoods in Manhattan and Brooklyn will not add 2-K seats next year. Mr. Mamdani has said that every 2-year-old in the city will have a spot by 2029.

Ms. Hochul has said that the state will pay for the first two years of the program. But it is not clear how the city will pay for future years when it could eventually cost $9 billion annually. Mr. Mamdani wants to raise personal income taxes on millionaires to pay for the program; Ms. Hochul has resisted the idea.

Ms. Hochul told reporters on Monday that the state was committed to continue paying for 2-K if she wins another term next month: “Universal means universal.”

When she was asked about raising taxes next year, Ms. Hochul did not rule it out. She said that the first two years of the child care program were covered by existing revenues.

“I want to make sure that this program endures, and we’ll do whatever it takes,” she said.

Ahead of the first day of school this year, many early childhood centers were alarmed that they had not received funding. The administration rushed to register contracts, extend loans and deliver basic school supplies.

Beyond the 2-K program, child care providers have also raised concerns that more than 25,000 low-income families have been stuck on wait lists to receive vouchers to help pay for child care. Mr. Mamdani recently announced that officials would move more than 10,000 children off the wait list.

Many families were excited to receive 2-K offers that would help them save more than $20,000 per year on child care. Other families were disappointed that they did not receive offers or declined them over concerns about quality or convenience.

The city already offers free preschool for 3- and 4-year-olds. Officials have sought to put the right number of seats in neighborhoods to match the demand. But that forecasting can be challenging. Some families prefer a preschool near work, while others are skeptical of enrolling in a city-run program at all.

Sean Geoghan, a father of two young children in the Inwood neighborhood of Manhattan, one of the first areas to receive 2-K seats, said he hoped to enroll his 4-month-old daughter in the program when she is 2. He said he was happy that most programs are full day and full year, instead of following the public school schedule.

“If child care ends before the work day ends, that simply does not work for our family,” he said.

Mr. Geoghan is on parental leave with his daughter and plans to send her to a home day care that costs about $2,400 per month. He said that his wife is Dominican and they are raising their children to be bilingual.

“We’d prefer to be in a home day care where they can be exposed to as much Spanish as possible,” he said.

The 2-K expansion mirrors the phased rollout of the 3-K program, which added additional school districts each year until it covered the entire city. That approach — rather than targeting smaller neighborhoods citywide with high economic hardship or high demand — helped create a surplus of vacant seats in some areas.

The neighborhoods targeted to receive 2-K seats next year have had mixed demand for 3-K and pre-K seats, according to a new report by the city’s Independent Budget Office. In a swath of the Bronx and Staten Island, demand for 3-K spots exceeded the supply during the 2024-25 school year, the report found. Yet there were unfilled pre-K seats that year in large areas of those boroughs.

The additional 10,000 seats could be filled regardless of local demand because some parents are willing to travel longer distances for a free spot. But when the program eventually expands citywide, some of the initial seats could be left empty if parents opt for more convenient sites.

The success of the 2-K expansion could depend on whether the city can sign up new child care providers. The pre-K and 3-K programs relied heavily on public schools and early childhood centers. The 2-K system is using centers and home day cares, which are popular in working-class communities.

Child care providers have long complained about inadequate funding from a contract that they signed with the city more than five years ago, before inflation soared during and after the Covid pandemic. (This year, the city agreed to raise the contract payments by 2 percent for early childhood centers.) In addition, providers are reimbursed by the city only after paying expenses out of pocket, and those reimbursements often take weeks or longer to arrive, leading to near-constant financial stress, operators said.

Mr. Mamdani has said that the city will hire 200 workers to stabilize the early childhood provider system, including the processing of payments.

Lori Cangiolose, the owner of the Playgroup Experience, an early childhood center on Staten Island, said that she would have no choice but to join the city-funded 2-K program. If she opted out, she said, parents would most likely leave for free care elsewhere.

Playgroup currently has 12 students enrolled in a private 2-K program paid for by parents, a vital revenue stream that has helped keep the center afloat when city reimbursements for its 3-K and pre-K programs have been late. But Ms. Cangiolose said she would lose that cushion if her business were almost entirely dependent on the city for revenue.

“We are giving up our businesses to the city,” Ms. Cangiolose said.

Lauren Melodia, the director of economic and fiscal policy at the Center for New York City Affairs at the New School, said that as the city expands its 2-K program, it should raise pay for child care workers, who often make near the minimum wage, and make it easier for home day cares to join.

“We want to make sure there are clear pathways for family care providers to participate,” she said.