New York City once sold a promise of free prekindergarten for all as an unusual benefit designed to make it far easier to raise children in this expensive city.

So as families worried over whether their 3-year-olds would have spots this fall, Mayor Eric Adams pledged last month that everyone would have “access” to a seat. But when the Education Department released offers this week, hundreds of families were left without a place after all, facing another potential year of child-care bills that often soar over $2,500 a month.

Every 4-year-old in New York is guaranteed a free preschool seat, and 3-year-olds were next in line for a universal program. But Mr. Adams canceled plans to expand that initiative because of empty slots in some neighborhoods, and he has cut millions of dollars from the program.

On Thursday, about 2,500 children did not receive a prekindergarten offer, leaving their parents in limbo. Many are still on huge waiting lists and scrambling to rethink their finances and future in the city.

Ben Lowe, a father in Prospect Heights, Brooklyn, did not receive a 3-K offer, and his daughter was behind dozens of other children on waiting lists for nearby programs. He said an extra year of paid child care could be a “devastating financial cost” for his family.

“I feel completely betrayed by this mayor,” Mr. Lowe said. “We don’t know how we’re going to solve this.”