By WILLIAM GLABERSON

Brooklyn judge refused yesterday to issue an injunction to stop the city from reducing the number of firefighters on some engines, clearing the way for the Bloomberg administration to make cuts that the firefighters' union says could cost lives.

While the cuts could have been made as soon as Thursday under the judge's order, Mayor Michael R. Bloomberg, after a late-afternoon meeting with Stephen J. Cassidy, the president of the Uniformed Firefighters Association, agreed yesterday to delay carrying out the plan for 30 days. A spokesman for the mayor said that he had agreed to the delay to give the union a chance to come up with alternatives to the cuts.

The battle over staffing on the engines