The Department of Education started doling out remedial funds to schools this week – money principals say they thought did not exist.

Last month, the city told principals there had been some budget cuts and said the first offer was final, individual principals and the principals union told The New York Sun.

But this week, the department’s chief financial officer, Bruce Feig, told the Sun that Regional Operation Centers can appeal budget allocations if they believe the funds are “insufficient to meet basic instructional needs.” If the department agrees, he said money could be allocated from a fund that is “held back centrally for this purpose.”

The deputy chancellor for finance and administration, Kathleen Grimm, said the money Mr. Feig said was “held back” is being distributed to schools that have more students than anticipated and to schools that would have to raise class sizes above levels allowed in the teachers’ union contract.

Earlier this month, principals said if their budgets didn’t change, they would have to fire teachers and raise class sizes.

Ms. Grimm said the department would make sure schools have enough money so they don’t have to break the contract.

Mr. Feig said the resources were being distributed through next week.

In addition, $500 million in new state and federal funds are now being electronically transferred into school budgets, according to this week’s “Principals’ Weekly.”

The late-arriving funds and the quiet budget appeals process are curing some of the most dire budget problems principals said they were facing.

“I think it’s wonderful,” the president of the principals union, Jill Levy, said of the new cash infusions.

But Ms. Levy, who was livid about the budget cuts earlier this month, said telling principals they had to make do with what they were handed and then later handing out extra funds created more “work,” “anxiety,” and “instability in the system” than was necessary.

A Jamaica principal, who spoke to the Sun on the condition of anonymity, said she had received about $20,000 new money this week. But she said she’s still almost $400,000 short of where she was last year.

“The only way I can get out of this trouble is if this appeal works,” she said.

Another principal, Patrick Burke, said he’s short about $300,000 and hasn’t heard of any appeals process. But he said, “There’s always a hope that someone’s going to pull something out of their hat by September.”