After saying he would stay out of the race, Eric Adams, the incoming mayor, has sought to install an ally. It was unclear whether he would succeed.When New York City’s mayor-elect, Eric Adams, said he was staying out of the combative race for City Council speaker, the declaration made sense: He still has the vast majority of his cabinet appointments to make, and an extraordinarily challenging job awaiting him in three weeks.And yet, skipping an important political contest — one that would decide whether the second-most powerful position in city government would be held by an ally of Mr. Adams — seemed like an almost novel notion in New York.And sure enough, in the past week or so Mr. Adams and his associates had spoken with council members on behalf of his preferred candidate, Francisco Moya of Queens, who was thought to have little pre-existing support among those who would have to choose him as their leader.The move, however, could end up backfiring.On Tuesday, four candidates for the leadership post agreed to quit the race and support a different contender, Adrienne Adams, a Queens councilwoman, bringing her closer to the 26 votes she needs to become speaker when the election takes place in January.“After much discussion and collaboration with my colleagues, I am honored to have received the necessary votes to become the next speaker of the New York City Council,” Ms. Adams said in a statement. “The incoming City Council will be beautifully diverse and wonderfully collaborative in so many ways.”But in an illustration of how complicated the jockeying has been, Mr. Moya also declared victory on Tuesday shortly after Ms. Adams’s announcement.“I am humbled to announce that our diverse coalition of council members and leaders from across New York City has collected a majority of votes to elect the next speaker of the Council,” he said on Twitter. “I look forward to leading this body into a brighter future for our great city.”Neither he nor Ms. Adams identified the council members they believe to be behind them.If Mr. Moya prevails, Mr. Adams would gain a trusted governing partner, helping to clear the way for him to execute his mayoral agenda.A victory by Ms. Adams, in contrast, would amount to a striking political defeat for a new mayor who had expended political capital to try to advance his choice, although Ms. Adams endorsed Mr. Adams in the Democratic primary and the two, who are not related, do not appear to have sharp ideological differences.Whatever the result, Mr. Adams has ended up at odds with key labor unions and council members at a vital moment in his transition, which has so far lagged behind that of his predecessor. The machinations threaten to create his first political headache as he seeks to fill out his administration.On Tuesday, Adrienne Adams of Queens gained the support of four council colleagues who had been vying to be speaker. Jeenah Moon for The New York TimesThe four council members who decided to drop out of the speaker’s contest and endorse Ms. Adams — Diana Ayala of East Harlem and the Bronx; Keith Powers of the East Side; Gale Brewer of the Upper West Side; and Justin Brannan of southern Brooklyn — did so after meeting with her on Sunday, The Daily News reported.The incoming City Council will be the most diverse in history, with women and members who identify as people of color in the majority. Some of the speaker candidates felt that it was important for a person who represents that diversity to lead the Council.“I recognize the moment we are in — that for the first time we have a women of color majority council, and I am proud to support Adrienne Adams as speaker of the Council who represents the body and this historic moment,” Ms. Ayala, one of three women of color in the race for speaker, said in a statement on Tuesday. Carlina Rivera of Manhattan remains in the race.(Later in the day, Ms. Ayala said the race had come to resemble a “telenovela.”)Should Mr. Adams fail in his first major political play as incoming mayor, he may have to grapple with lingering bitterness on the part of the City Council and some labor leaders upset by his team’s efforts to override their will.In interviews over the past several days, some council members have expressed concern about the advisers who worked with Mr. Adams to support Mr. Moya in the first place. These members also questioned why Mr. Adams has appeared unwilling to back down in the face of substantial resistance from the Council and several influential labor unions, including 32BJ, which represents building service workers, and DC37, the city’s largest public employee union, that wanted a more collaborative process.The battle over who will be the next council speaker has consumed New York politics at every level in recent days. Labor leaders, county officials and other party leaders have all sought to square the preferences of the incoming mayor, who is at the height of his political power and enjoying reservoirs of good will, with an incoming class that includes independent-minded members. Some of them have deep reservations about Mr. Moya, although certainly he now has public endorsers as well.Mr. Moya is Latino and, at a moment when no citywide office is held by a Latino, a number of his supporters have cited his background in making the case for his candidacy.Mr. Moya’s foes have taken to using a “Star Wars” term to characterize their opposition, referring to themselves as the Rebel Alliance.Sarah Stier/Getty Images For QbfcThe race has even created divisions within Mr. Adams’s own coalition — so much so that some council members and others involved in the process who oppose Mr. Moya have referred to themselves as the Rebel Alliance, a “Star Wars” reference.“It is the most senselessly divisive speaker’s race I have ever seen,” said Representative Ritchie Torres, a Democrat from the Bronx and a former council member, speaking broadly of the upheaval. “A wedge has been driven through the congressional delegation, the county organizations, the council and the labor movement.”But Staten Island Councilman Joseph Borelli, who leads the council’s five-member Republican caucus, said the maneuvering was not unusual.“People can’t remember four years ago and they can’t remember four years before that,” said Mr. Borelli, who has yet to publicly declare his allegiance in the race, but is believed to support Mr. Moya. “And there’s always this level of horse-trading and ebbing and flowing of momentum.”Representative Grace Meng, a Queens Democrat who is close to three members of the incoming council who are thought to be undecided, has been urging union officials, Mr. Adams and county party leaders to hold a meeting to sort out their differences.Tiffany Cabán, a newly elected Queens council member and a democratic socialist, said it was important that the next speaker be able to “democratize” the City Council and help members craft an agenda based on the needs of the communities they represent.“There are a few things that are incredibly important in the next speaker,” Ms. Cabán said. “First and foremost is that the mayor isn’t picking the speaker. The best way to move forward is with an independent body that provides checks and balances.”Despite multiple people describing Mr. Adams having direct conversations about the race with council members, he continues to maintain that he is taking a hands-off approach.“I stated it from the beginning, that I was not going to be heavy-handed in this race,” he said, during a TV interview on Tuesday morning. “People call me, they call me all the time, and they say, ‘Eric what are your thoughts?’ I give them an analysis of the people who are mentioned in the race, to give my input. I’m looking for a partner. The speaker does not work for me.” More