The inflation-battered Cambridge school budget emerged from the school committee last week and headed for city council, arriving two hours before the final deadline and immediately sinking into a new bog of troubles.

An all-night session a week ago settled most of the budget issues--the deepest cuts recommended by Superintendent William Lannon were ignored, but only at the expense of nickel and dime reductions in other programs.

Some cuts remained, though, that troubled school committee members. One of the most obvious--the elimination of freshman and sophomore interscholastic athletics--will be debated again in a few weeks when school board members try to find dollars from elsewhere in the athletic department budget to restore the program.

But hardly had the budget reached City Hall before a new controversy developed. Not included in the tally was more than $6 million in funding for teacher's health insurance and other programs.

Traditionally the funding has been part of the city's budget. This year, however, City Manager James L. Sullivan announced, that would change; "I want this for psychological reasons--so that the committee will know exactly how much it costs to run the schools," Lannon said.

School committee members, however, fear the political effects of the transfer and also the problems insurance rate hikes might cause in the future. "I don't want educational programs subject to the whims of the rate-setting agencies," Glen Koocher '71, school committee member insisted.

The city manager apparently has the council votes right now to keep the $6 million out of his budget. If that support holds, the school committee will probably be forced to absorb the costs and return to the city council with a supplemental budget later in the year.