(Repeats to widen distribution)

NEW YORK, April 8 (Reuters) - New York Mayor Michael Bloomberg has asked city agencies to cut another 7,000 jobs as the city's finances continue to deteriorate in the recession.

In a letter to agency heads, Mark Page, the head of the city's Office of Management and Budget, said that tax revenues continue to fall short of projections, forcing the city to further trim costs to balance its budget.

The OMB is open to ideas for cuts, but "given the efficiencies you have already achieved, this next step would most likely rely heavily on additional headcount reductions, whether through attrition, or, as is more likely, through layoffs," the letter said.

The new round of cuts aims to reduce spending for fiscal 2010, which begins July 1, by another $350 million, Page said.

The cuts come on top of the headcount reductions of about 9,000 positions that were included in the January preliminary budget for 2010. Those cuts included 1,300 layoffs.

Union leaders weighing in on the letter said cutting staff has proved to be a failed strategy in the short- and long term.

"The OMB statement, besides outlining fiscal problems, is unfortunately an effort to negotiate with Albany and the unions in public," said Harry Nespoli, chairperson of the Municipal Labor Committee, in a statement.

Agency heads have a deadline of noon on April 13 to reply to the letter.