Seeking to Succeed Bloomberg, While Keeping Him at a Distance Mayor Michael R. Bloomberg at City Hall last year with Christine C. Quinn, the City Council speaker. An endorsement of Ms. Quinn by the mayor was once treated as a foregone conclusion. Michael R. Bloomberg is a data-loving, health-enforcing nonpartisan who defined what it meant to be a mayor over the past 12 years, mentoring big-city chief executives from New Orleans to Philadelphia. But closer to home, the major candidates vying to succeed him at City Hall in New York are not pleading for his pointers, or pressing for a place in his photo ops: none of them, Democrat or Republican, have even asked for his political blessing in the primaries next week. They instead describe polls showing that Mr. Bloomberg's endorsement, once deemed a coveted prize, would now dampen their appeal to the party faithful, not burnish it, and they fret over lashing themselves to his divisive policies on policing, not to mention his soured relationship with municipal unions. "A double-edged sword - or worse," is how Bill de Blasio, the front-runner in the Democratic field, characterized support from Mr. Bloomberg. Christine C. Quinn, long treated as the mayor's heir apparent, was more diplomatic, but no less conflicted: "You know," she said, "it's a complicated question." It is a humbling and alien experience for the relatively popular mogul turned mayor, who is unpracticed in humility: in the race to lead a post-Bloomberg New York City, there is profound wariness of being viewed as, well, too much like Mr. Bloomberg. The arms-length approach to the powerful mayor demonstrates how thoroughly the campaign has been reframed by the fiery, anti-Bloomberg message and rhetoric of a single candidate - Mr. de Blasio, the city's public advocate. Throughout the summer, Mr. de Blasio has portrayed the current administration, and its allies, as protectors of an unacceptable status quo that coddles the rich and overlooks the poor, elevates Manhattan and ignores its neighboring boroughs. In the process, Mr. de Blasio has forced his rivals to recalibrate how they talk about Mr. Bloomberg and the future of New York. "He has shown that you can succeed by complaining about Bloomberg," said Kenneth Sherrill, a professor emeritus of political science at Hunter College in Manhattan. That was something that most candidates were afraid to do. Now that the critique of the Bloomberg era has become so central to Mr. de Blasio's surging candidacy, Mr. Sherrill said, rival campaigns are struggling to match it. They are all playing catch-up - and it's obvious. Not even candidates from the Republican Party, which propelled Mr. Bloomberg into the mayoralty, are immune to the anxiety. A poll commissioned by a Republican mayoral campaign found that two-thirds of the party's rank-and-file voters had a negative view of Mr. Bloomberg. "He is not an asset in a G.O.P. primary," said an operative who had seen the results of the poll, speaking on the condition of anonymity to shield the identity of the candidate behind the survey. Mr. Bloomberg has a somewhat tortured relationship with the two major political parties: once a Democrat, he ran for mayor as a Republican, then became an independent. That journey has engendered bitterness in each scorned camp, and earned him little love from the partisan voters who may settle the Sept. 10 primaries. In the unsettled relationship between the candidates and the mayor, it turns out that the apprehension is mutual. Mr. Bloomberg has openly despaired about the aptitude and readiness of this year's field, so much so that he has repeatedly tried, but failed, to recruit what he felt were better-qualified outsiders, like Hillary Rodham Clinton and Raymond W. Kelly, to jump into the race. The curious result is a primary campaign in which the candidates are not seeking Mr. Bloomberg's seal of approval and he has no plans to bestow it, for now anyway, people close to him said. City Hall, always attuned to the mayor's legacy, is highly sensitive about the anti-Bloomberg tone of the campaign. A spokesman for Mr. Bloomberg declined to comment for this article. Representatives for the five major Democratic candidates and the two leading Republican campaigns all confirmed that they had not sought Mr. Bloomberg's endorsement in their parties" nomination contests. Of course, those political calculations could be scrambled, for both the mayor and his would-be successors, after Tuesday's primaries, depending on the matchups in the general election. There is scant enthusiasm inside the Bloomberg administration for Mr. de Blasio, a staunch liberal; but there is a natural kinship with Joseph J. Lhota, a moderate Republican contender who has spent much of his career in the private sector. In a series of interviews with The New York Times over the summer, those seeking the mayoralty spoke candidly and critically about Mr. Bloomberg's legacy and his role in the campaign. William C. Thompson Jr., a Democratic candidate and a former city comptroller, faulted the mayor for "not listening to the people," pointing to Mr. Bloomberg's insistence last year that the city hold the New York City Marathon days after Hurricane Sandy cut a violent path across the city. The race was eventually canceled.