PRIMARY RACES: still several turns before stretch run; In Illinois, Kennedy hopes Jane can deliver

Chicago's feisty Mayor Jane Byrne, then a fresh new figure on the national political scene, was at Sen. Edward M. Kennedy's side last fall when he launched his presidential quest at Boston's Franeuil Hall.

Now, with a week to go before crucial Illinois primary, Mrs. Byrne still is battling hard for the senator -- and is one of the few big city mayors beside him.

But whether Mrs. Byrne is actually helping or hurting the senator appears as hard to judge as the Illinois race itself.

Chicago has been in turmoil under Mayor Byrne's rule. And Kennedy Illinois spokesman Terry Michael acknowledges that "the firemen's strike, and before that the teachers' strike, blocked out media attention to the presidential race."

To Mr. Kennedy, possibly leading in Chicago but trailing Mr. Carter statewide , the greatest remaining campaign problem is "use of the media to change minds," Mr. Michael says. Settlement of the firemen's strike March 8 should help swing attention in this heavily Democratic city back to the struggle for the party's presidential nomination.

Mrs. Byrn's power has helped, Kennedy people say. She squashed efforts to put up neutral, uncommitted delegate slates. Legions of Chicago and Cook County patronage workers are distributing Kennedy campaign literature, and will help turn out the vote.

And Mayor Byrne has helped with fund raising. "We raised $80,000 with her help last Friday," said Mr. Michael. "We've raised $500,000 in Illinois, more than our share of national campaign money."

But others see the Byrne influence differently.

"I think she's hurting him," says Milton Rakove, University of Illinois analyst of Chicago political affairs. "She's arousing so much animosity.

"Byrne can't deliver the machine vote the way [the late Mayor Richard] Daley did. She has literally brought down the Daley system in Chicago. She's cleaned out the whole Daley bureaucracy. Her own city staff is incredibly weak."

Mayor Byrne's rivals, says Mr. Rakove, are "marking their time, watching the show, waiting for her to fall apart.

He adds: "The precinct captains in the city aren't that happy with Kennedy. They had a hard time getting the petitious signed. They will go through the motions to keep Byrne out of their hair.

"In 10 or 12 wards this time, the incumbent ward leaders are up for re-election," Mr. Rakove points out. He feels the captains may concentrate on the ward race or the race for state's (county) attorney race.

President Carter appears to be having his troubles in Illinois, too. He seemed a "shooin" until his United Nations turnaround on the Israeli settlements. "He blew the Jewish vote," observers here say.

But the Chicago Jewish vote, influential in the north side of the city and in the suburbs, may turn to Republican John B. Anderson rather than to Mr. Kennedy. Many of the city's strongest Jewish leaders are talking about supporting Mr. Anderson "as a way to register our feelings with Carter."

Senator Kennedy does have other support in Illinois besides that of Mrs. Byrne. He still seems likely to get the backing of Chicago's black and Hispanic populations, whom the mayor has failed to champion. Together, blacks and Hispanics make up more than half of Chicago's 3 million population.

Also, coal miners in southern Illinois appear to be backing the Massachusetts senator, as will the large black population in East St. Louis, Ill.

"What we have going gor us in Illinois are the elements of the Democratic constituency: organized labor, large minority groups, and the city," Mr. Michael says.

Cutting the other way, "Chappaquiddick is a factor," says a Kennedy spokesman , "but nobody knows how much a factor."