Starting in September, some children attending taxpayer-funded prekindergarten programs in New York City may have school on Christmas and Thanksgiving. And their school day may be interrupted by a period for religious prayers.

The new rules are a result of a compromise by Mayor Bill de Blasio’s administration that is intended to increase participation by Orthodox Jewish and other private schools in his signature initiative.

Most of New York’s new prekindergarten programs are run by private organizations, including religious schools, that are compensated by the city. But many Orthodox schools have stayed away because they feel the city’s requirement of 6 hours 20 minutes per day of secular instruction does not leave them enough time for religious schooling.

Discussions between City Hall and Orthodox Jewish and other private schools have continued since September. The mayor is seeking to reach his goal of some 70,000 children in full-day prekindergarten by the fall; 53,000 are enrolled now.