Candidates vied for the backing of the influential teachers’ union and other players, and one contender made a journey to Minneapolis.New York City’s mayoral contest is indeed beginning to pick up steam.A handful of endorsements were issued, and two more influential ones — from the United Federation of Teachers and the Working Families Party — may soon follow.Candidates also fanned out across the city and beyond. Andrew Yang, the former presidential candidate, appeared with the city’s celebrity-of-the-moment: Huge Ma, the creator of the TurboVax website that makes it easier to schedule a vaccine appointment.Maya Wiley, the former MSNBC analyst and counsel to Mayor Bill de Blasio, did a sprightly double Dutch jig on a sunny day in the Bronx. And Raymond J. McGuire, a former Citigroup executive, visited Minneapolis to pray for justice outside the courthouse where the former police officer Derek Chauvin is on trial in the killing of George Floyd.Here’s what you need to know about the race:Blue-collar support vs. a ‘change-maker’Maya Wiley, left, was endorsed by Representative Yvette Clarke, a Brooklyn Democrat, who said that Ms. Wiley was the “change-maker this moment calls for.”Gabriela Bhaskar for The New York TimesTwo Democratic members of Congress made mayoral endorsements last week: Tom Suozzi, a moderate who represents parts of Queens and Long Island, and Yvette Clarke from Brooklyn, who has one of the most liberal voting records in Congress.Mr. Suozzi backed Eric Adams, the Brooklyn borough president; Ms. Clarke supported Ms. Wiley.Mr. Suozzi emphasized Mr. Adams’s focus on “blue-collar workers” and the fact that Mr. Adams “led while others fled” — a reference to Mr. Yang, who spent portions of the pandemic at a second home in the Hudson Valley with his family, leaving his apartment in Manhattan.Ms. Clarke said Ms. Wiley is “the change-maker this moment calls for” and would bring “competence and compassion to City Hall.” She said Ms. Wiley was one of the first top-tier mayoral candidates who “embodies the feminine” — apparently ignoring or discounting past Democratic hopefuls like Ruth Messinger, the first woman to win the party’s mayoral nomination, or Christine Quinn and Bella Abzug, who both lost primary elections for the Democratic nomination, as well as a Republican candidate, Nicole Malliotakis, now a member of Congress.Ms. Clarke later clarified that she did not intend to “diminish or erase” other women running for mayor this year or in past elections.Scott M. Stringer, the city comptroller, received an endorsement from the union that represents public school principals. Mr. Yang received support from Matthew W. Daus, the city’s former taxi commissioner whose role in the current taxi crisis has been scrutinized.Mr. Yang also landed an appearance, but not quite an endorsement, from Mr. Ma, sometimes known as “Vax Daddy,” at a news conference outside a vaccination site in Washington Heights, and said he wanted to hire him in a Yang administration. Mr. Ma said he was not ready to back a candidate quite yet.“The only thing I am ready to endorse is more protected bike lanes,” he said.An unorthodox way to seek an endorsementOne of the last major unclaimed endorsements should be decided this month, and the top four candidates — Mr. Stringer, Ms. Wiley, Mr. Adams and Mr. Yang — made their cases on Wednesday at a forum hosted by the United Federation of Teachers.Of the four, Mr. Yang seemed to be the most willing to diverge from union orthodoxy, seemingly hurting his chances for the endorsement.He was the only candidate to unequivocally say he wanted to retain mayoral control of city schools as is, contrary to union efforts to weaken mayoral control. He was also the only candidate to admit to not reading the union’s five-point plan for reopening schools in September.But Mr. Yang, who has one son in public school and another in private school, said that he no longer entirely blames the union for reopening delays during the pandemic.He said he now understood that Mayor Bill de Blasio was to blame, too, relating a conversation he had with Michael Mulgrew, the union’s president and the forum’s moderator.“You conveyed to me that it’s been a failure of leadership on the part of the mayor and that the teachers need a partner who’s committed to reopening the schools in a responsible way that protects teachers and makes everyone feel safe and secure,” Mr. Yang said. “I agree that the mayor has failed the teachers and public school parents like me.”The comments represented a departure from what he told Politico in March, when he said that “the U.F.T. has been a significant reason why our schools have been slow to open.”They also seemed to offend the current mayor.“I don’t know what Andrew Yang is talking about now,” said Bill Neidhardt, a spokesman for the mayor. “Mayor de Blasio was the only big-city mayor to open up schools.”Campaign trail extends to MinneapolisRaymond J. McGuire visited Minneapolis to attend a prayer service outside the courthouse where the former police officer Derek Chauvin is on trial in the death of George Floyd.Stephen Maturen/Getty ImagesMr. McGuire is a moderate on policing. He served on the New York City Police Foundation, a nonprofit that raises money for the Police Department, and has not called for police to be defunded as some of his fellow candidates have.So his decision to travel last week to Minneapolis, where the officer who knelt on George Floyd’s neck for almost 10 minutes is on trial for murder and manslaughter, seemed to convey a political message.Mr. McGuire was there with Gwen Carr, the mother of Eric Garner, who died after a New York Police Department officer placed him in a chokehold in 2014 on Staten Island; the Rev. Al Sharpton; and a former New York State governor, David Paterson. The group met privately with Mr. Floyd’s family.Mr. Sharpton said that Mr. McGuire called and asked if he could join him in Minneapolis, where he has traveled regularly to support Mr. Floyd’s family. He said Mr. Garner’s death showed New York is not immune from such tragedies.“This is not an isolated issue or a question or whether it could happen here; it did happen here,” Mr. Sharpton said.Mr. Sharpton has yet to endorse in the race for mayor, but said that he viewed Mr. McGuire’s interest in Mr. Floyd’s trial as a good sign. “He was the only one who asked to go, and that speaks for itself,” he said.Candidates for mayor should take note of the Minneapolis police chief’s testimony that his officer had violated departmental rules in Mr. Floyd’s death, Mr. Sharpton said, adding that the next mayor should dismantle the so-called blue wall of silence.Mr. McGuire, who said that the “evidence is incontrovertible” that Mr. Floyd’s death was a criminal act, said that policing in New York could be “fixed with the right leadership.”“We want better policing,” Mr. McGuire said. “You go to the neighborhood, and people aren’t talking about defund.”70 plans in 70 daysShaun Donovan, a former federal housing secretary, will announce a “70 Plans in 70 Days” campaign, highlighting one idea every day until Primary Day.David Dee Delgado/Getty ImagesShaun Donovan, the former federal housing secretary, is trying to distinguish himself in the crowded field as the policy wonk who has the best proposals to improve the city.Mr. Donovan will announce his “70 Plans in 70 Days” campaign this week and highlight one idea every day until Primary Day.“By June 22, I am confident that every New York City voter will know who the most qualified person is to lead our city through this crisis, and who has the actual plans to get the job done,” Mr. Donovan said.His proposals include equity bonds, a government-funded savings account for every child; 15-minute neighborhoods, where every resident has access to a good school, rapid transit and a beautiful park within 15 minutes of their home; and allowing noncitizens to vote in city elections.Mr. Donovan has a pile of cash to help get the word out: His father, Michael Donovan, has given $2 million to a super PAC for his son.Mr. Donovan has been lagging in the polls, and his campaign took aim at Mr. Yang last week. It joked that Mr. Donovan would not climb atop salt piles when visiting a sanitation facility like Mr. Yang did.One of Mr. Yang’s campaign managers hit back on Twitter: “Will his dad not let him climb on the salt?”Andrew Yang is writing a bookIn February, Mr. Yang made headlines for skipping a forum with Muslim groups on the same day that he spoke on a podcast hosted by Sam Harris, who has made incendiary remarks about Islam.Less noticed at the time: At the conclusion of Mr. Yang’s nearly hourlong appearance on the podcast, he indicated that he was writing a book — one that apparently deals in part with the new system that will be used in the mayoral election, ranked-choice voting.He said on the podcast that the book was slated to come out in the late summer.On Sunday, Mr. Yang’s campaign confirmed that Mr. Yang, already an author, indeed had another book coming, to be published by Random House, and that he finished a draft at the end of last year.“Andrew is solely focused on the mayoral campaign and will not be publishing this book until afterwards,” said Eric Soufer, a spokesman for Mr. Yang. “The book will be about Andrew’s experience in the presidential campaign, along with his vision for decreasing polarization, increasing turnout and improving the health of our democracy.”The book, he said, ends before Mr. Yang’s run for mayor. 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