New York Gov. Andrew Cuomo and state lawmakers missed the April 1 deadline for passing the state budget by the start of the new fiscal year and were still far apart on policy issues on Saturday.

Mr. Cuomo issued legislators a new deadline: Pass the budget by midnight on Sunday or he will pass an extender of the current budget instead.

Mr. Cuomo’s extender would last until May 21, when the U.S. Congress is expected to pass its budget resolution.

The governor said this would have the benefit of giving lawmakers more clarity about potential cuts to New York from a new Republican congress and president.

In a statement minutes after midnight on Saturday, Mr. Cuomo, a Democrat, said the basic outlines of the roughly $160 billion spending plan for the new fiscal year were resolved, but policy issues that lawmakers intend to pass with the budget weren't.

Two obstacles were a measure to remove minors from the criminal-justice system and a replacement to 421-a, the expired law that gave real-estate developers tax breaks in exchange for including affordable housing in their real-estate portfolios.

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