After almost a dozen public meetings concerning 10 community center-based preschool and after-school programs slated to lose teacher funding, Mayor Thomas Menino has granted yet another concession. The five preschool programs under threat, including the Murphy Community Center, will now be funded until at least the end of the next fiscal year, which ends July 2010.

“For the most part, what we heard all along was ‘We need more time,’” said Boston Centers for Youth and Families director Daphne Griffin on Tuesday. “I checked in with [Menino] yesterday. The mayor has offered, on the preschool programs, to extend funding through the end of the fiscal year. And that’s really what we heard from the councils.”

The original plan was to shift the burden of funding the teacher’s pay over to the respective site councils, which are independent non-profits. That plan is still in effect for the five after-school programs, including the Marshall, which will lose their teacher funding from the city as of September. Griffin has promised the BCYF will work with each council to apply for grants to keep the programs running afterward.

Each community center has a council, though some, like the Holland, Marshall and Cleveland, are gathered in clusters under one council. Menino’s move to fund the preschools for another year follows an emergency City Council hearing held last week by councillors Sam Yoon and John Tobin, and anticipates an ordinance to fully restore the preschool and after-school funding that the two were planning to put forth for a Council vote this week.

“The BCYF staff didn’t even get an offer for a wage freeze, and they [the union] showed how they could achieve the necessary budget cuts without a pink slip,” said Yoon, relating testimony he heard from Susana Segat, head of SEIU 888, which represents the teachers. Segat did not return calls for comment this week.

“It’s crazy,” continued Yoon. “As far I’m concerned there wasn’t a clear or compelling reason.”

Griffin has explained that 32 other community centers in the BCYF system run and pay for their own preschool and after-school programs, and the administration wants to make that pattern the norm. Making that happen would also save the city money on wages and benefits. The preschools are paid for with a combination funds from the city, from vouchers, and from rates paid by parents, she said. Some of the after-schools are funded by state Early Education and Care grants, and the Marshall’s council, for one, is applying for the same type of grant to continue its program.

The Perkins, the third Dorchester center affected by the cuts, plans to shift the control of its after-school program to an outside non-profit.

Griffin said the cuts do not indicate a change in her wider effort to affect some positive changes in the governance structures of some centers.

“That hasn’t gone away,” she said. “We’ve had an infusion of new people at the Murphy, and a great collaboration is at work there… That doesn’t change the overarching challenge that the governance structures present.”

Though there are some shining examples of centers in the BCYF system that are well run by their councils, such as the Sherbourne in Roxbury, others have been derided as ‘little fiefdoms’ that are ill-managed or stagnant, with little community programming. Emily Loska, an attendee of the Marshall’s after school when she was a child, testified to this effect at the council hearing last week.

“They don’t answer to BCYF, they don’t answer to the public, they’re getting money in the name of the neighborhood and the neighborhood doesn’t know it’s there,” she said in a phone interview Monday.

Loska’s mother Linda Loska was a founding member of the Holland Council, which oversees programming at the Dorchester Cluster of community centers.