Gov. Cuomo’s new state budget shifts at least $56 million in costs to New York City — but also includes more than $400 million in funding increases for the city, say governor’s aides.

Reimbursements to city agencies for placement of special-education students and foster-care kids is being slashed by $42 million.

Other cuts include $11 million for city public-health programs. Increases include $295 million more for education, $129 million extra for Medicaid and $55 million for other programs.

Mayor de Blasio’s office said City Hall said is still reviewing the budget numbers after being briefed on “broad strokes.”