Andrew Yang made a big splash last week as he entered the mayor’s race, adding a jolt of energy to a campaign season that has been relatively staid and polite until now.

Other campaigns pounced on Mr. Yang, questioning his authenticity as a New Yorker and his commitment to the city. While their digs highlighted some of his weaknesses, they also illustrated how the candidates view Mr. Yang as a threat.

The campaigns also released their fund-raising figures last week, showing which candidates are in the strongest financial position, while a former Wall Street executive known for a #MeToo complaint entered the lesser-known Republican field.

Here are some key developments in the race:

The knives are out for Yang

Even before Mr. Yang entered the race, he had already faced social media ridicule for a remark he made to The New York Times explaining his decision to leave New York City for his weekend home in the Hudson Valley early in the pandemic.

That was before the bodega incident.

A day after Mr. Yang held an in-person campaign launch in Morningside Heights, he posted a video on Twitter about his love for bodegas — a safe stance that few would challenge. But Mr. Yang recorded the video in a spacious, glistening store that few New Yorkers would consider a bodega.

The video brought Mr. Yang more ridicule — and 3.7 million views by Sunday afternoon.

Rival campaigns took other swipes at him. After Mr. Yang finished a walking tour of the Brownsville neighborhood in Brooklyn, the campaign of Eric Adams, the Brooklyn borough president, said: “Eric doesn’t need a tour of Brownsville. He was born there.”

The campaign manager for Maya Wiley, a former counsel for Mayor Bill de Blasio, knocked Mr. Yang’s swerve from the presidential campaign trail to the New York mayor’s race: “Maya is running — not as a backup plan — but because she’s dedicated her entire life to improving, empowering and lifting up New Yorkers.”

Mr. Stringer’s campaign spokesman, Tyrone Stevens, also took a dig: “We welcome Andrew Yang to the mayor’s race — and to New York City.”

Could Yang’s entry be divinely inspired?

The choice of music to accompany a candidate’s official launch or acceptance speech is usually a calculated decision. “Don’t Stop” by Fleetwood Mac was Bill Clinton’s campaign theme song in 1992; Lorde’s “Royals” prefaced Mr. de Blasio’s victory speech in 2013.

Mr. Yang arrived at his launch event in Morningside Park in Manhattan dancing to the song “God’s Plan” by Drake, which features the lyrics: “They wishin’ on me/Bad things.”

Indeed, Mr. Yang faced a barrage of questions from journalists over why he left the city during the pandemic and why he had not voted in local elections. One key question is whether Mr. Yang views the job as a steppingstone to run for national office again — like Mr. de Blasio who received criticism for his lackluster bid for president in 2019 and several trips to Iowa.

Asked by The New York Times if he would commit to not running for president while serving as mayor, Mr. Yang declined to do so. But he said being mayor of New York would be the job of a lifetime.

“New Yorkers have nothing to worry about,” he said.

Mr. Yang made a suggestion: The city should take control of the subway away from the state. There’s only one obstacle: Gov. Andrew M. Cuomo, who has seized near total control of the transit agency and is not known for ceding power.

“Who knows? Maybe he’ll be happy to have the city take it off his hands,” Mr. Yang told reporters gathered on a subway platform, who laughed incredulously at the thought.

He spent his first day on the campaign trail crossing four of the city’s five boroughs (sorry, Staten Island). On NY1’s “Inside City Hall” that evening, Mr. Yang disappointed some by saying the city might not be able to close the jail at Rikers Island by the year 2027.

“Rikers Island should be closed, but we need to be flexible on the timeline,” he said.

A rising star’s endorsement stirs resentment

Mr. Yang touted a key endorsement as he hit the trail: Representative Ritchie Torres of the Bronx, a rising star in the Democratic Party who helped to counter criticism that Mr. Yang was out of touch with the city.

Mr. Torres and Mondaire Jones are the first openly gay Black men to serve in Congress, and Mr. Torres was being courted by several campaigns. He had met or had conversations with Ms. Wiley, Mr. Adams, Mr. Stringer, Raymond J. McGuire and Shaun Donovan, a former housing secretary under President Barack Obama.

Mr. Torres said he gave the losing campaigns a heads-up about his decision even while he was engrossed with the vote to impeach President Trump.

“No mayoral candidate endorsed me in my race,” Mr. Torres said. “I did not owe anyone anything.”

Mr. Torres said Mr. Yang’s endorsement of a universal basic income would be a victory for the South Bronx district he represents, one of the poorest in the nation. He said that he also likes the fact that Mr. Yang is not a part of the city’s political establishment.

The endorsement allows Mr. Torres to align himself with a fellow moderate progressive. If Mr. Yang wins, it would boost Mr. Torres’s standing while giving him a powerful ally in City Hall.

Asked about the response to his decision, Mr. Torres said: “Eric Adams was gracious, most were disappointed and one campaign in particular was hostile.”

Several people who were familiar with the discussions said that the McGuire campaign was the one that responded with hostility. Mr. Torres met with Mr. McGuire, a former Wall Street executive, at an event in the Hamptons over the summer, and his campaign believed it had the inside track.

Mr. McGuire’s campaign denied being upset about the snub.

“Ray is not a politician and doesn’t hold grudges,” his spokeswoman, Lupé Todd-Medina, said. “He looks forward to working with the congressman when he is mayor.”

Kathryn Garcia has fans but is short on funds

Many officials who have worked in and around city government think highly of Kathryn Garcia, the city’s former sanitation commissioner, who is running as a proven manager capable of leading the city’s recovery from the pandemic. But she is falling behind in the money race.

Ms. Garcia has raised about $300,000 and failed to qualify for public matching funds.

Still, the recent filings revealed that Ms. Garcia received campaign donations from a number of high-powered New Yorkers, including Joseph J. Lhota, the former head of the Metropolitan Transportation Authority who ran as Republican against Mr. de Blasio in 2013; Polly Trottenberg, the city’s former transportation commissioner; and Kathryn Wylde, the leader of a prominent business group. Ms. Wylde also donated to Mr. McGuire, who is a favorite among Wall Street donors.

Monika Hansen, Ms. Garcia’s campaign manager, said that many city employees back her bid.

“Kathryn has the support of the doers in New York City government at every rank,” she said.

A lesser-known candidate, Zachary Iscol, a nonprofit leader and former Marine, has raised nearly $750,000 and said he expects to qualify for matching funds soon.

Another candidate who worked in Mr. de Blasio’s administration is struggling: Loree Sutton, a former veteran affairs commissioner who has $398 on hand and $6,000 in outstanding liabilities. She said her campaign has experienced some bumps but that she is reorganizing and “is in this race and in it to win it.”

A former Wall Street executive joins as a Republican

The Democratic primary in June is expected to decide the mayor’s race, with registered Democrats far outnumbering Republicans in New York City. But there is also a Republican primary in June, and a new candidate entered the race last week: Sara Tirschwell, a former Wall Street executive who once filed a #MeToo complaint against her boss.

In an interview, Ms. Tirschwell touted her experience as a single mother and a moderate Republican with liberal social views. She highlighted her “managerial competence” as a rare woman who rose to high positions at financial firms.

“I think there is a need for a moderate in this race, and it’s not clear that a moderate is going to survive a Democratic primary in New York City,” she said.

Ms. Tirschwell, who grew up in Texas, echoed the complaints of many Republicans — and some Democrats — that “Bill de Blasio is probably the worst mayor in our lifetime.” But she did not want to discuss the recent violence in Washington or Mr. Trump’s impeachment.

“This race is about New York, and it’s about New Yorkers and the crisis that this city faces, and that is what my campaign is focused on,” she said.

Other names that have been floated in the Republican primary: John Catsimatidis, the billionaire owner of the Gristedes grocery store chain; Fernando Mateo, an advocate for livery cabdrivers who was linked to a scandal over Mr. de Blasio’s fund-raising; and Curtis Sliwa, the founder of the Guardian Angels.