With the presidential race and the Senate elections in Georgia decided, the mayor’s race in New York City now takes center stage, with money in the starring role.

The candidates are scrambling to secure donations ahead of a major fund-raising deadline this week, trying to stand out in a crowded field that will likely soon include Andrew Yang, the ex-presidential candidate.

An indoor fund-raiser was held over the weekend for Eric Adams, the Brooklyn borough president, even though he has received criticism for raising money indoors during the pandemic. A Marvel superhero actor has backed Maya Wiley, a former top counsel to Mayor Bill de Blasio, who seems in danger of not qualifying for matching public funds.

And at the latest mayoral forum ahead of the Democratic primary in June, the candidates sought to pitch an ambitious idea for the city and to position themselves as authentic New Yorkers.

Here are five highlights from the race last week:

Adams skips an indoor fund-raiser held for him

The city might be in the midst of a second wave of the coronavirus and facing the prospect of a more transmissible variant, but that has not stopped some in-person mayoral fund-raisers.

On Saturday, weeks after Mr. Adams prompted an outcry for continuing to fund-raise indoors, another gathering of his donors took place — this time across the New York City border in Great Neck, Long Island, where indoor dining is still allowed. Mr. Adams did not attend the event, at an Asian fusion restaurant.

Peter Koo, a city councilman who represents Flushing, Queens, hosted the event. To entice his friends to attend, he played up how Mr. Adams believes in “law and order” and wants to retain a controversial exam that determines entrance to specialized high schools, Mr. Koo wrote in a text message to friends.

“His administration will have a diversified cabinet,” Mr. Koo also wrote, according to a copy of the message shared with The New York Times. “If elected, he said he will appoint me as one of the deputy mayors.”

In an interview on Sunday, Mr. Koo said it was a mistake to include that last line and that it wasn’t true.

“I am at retirement age,” said Mr. Koo, 68. “I don’t need a job. I said that just to impress on my friends, do this, help support a mayor.”

Mr. Adams is officially on the same page.

“That is categorically false,” said Evan Thies, Mr. Adams’s spokesman. “There have been no conversations about staff roles, nor has anyone been offered a role. Appointments will be based on only one standard: ability.”

He also defended the use of indoor fund-raisers, which many campaigns have avoided.

“We ask all fund-raiser hosts to follow health and safety guidelines specifically outlined by the campaign and in compliance with the law,” Mr. Thies said.

Anyone with a big idea, please raise your hand

For all the criticism of Mr. de Blasio’s tenure, he was successful in delivering a centerpiece of his platform: prekindergarten for all.

At a mayoral forum hosted by Uptown Community Democrats last week, candidates were asked to name one big issue that they would “completely resolve” if given two terms.

Ms. Wiley and Shaun Donovan, a former housing secretary under President Obama, both committed to ending street homelessness.

“We need to put people in housing first — not spend $2 billion for a shelter system that frankly people are sleeping on the streets to avoid,” Ms. Wiley said.

Mr. Adams has proposed a bold idea that also affects school instruction, possibly to the chagrin of students: extending school year-round by shortening the summer break and adding shorter breaks during the year. He said he wants to keep students engaged over the summer, especially as families recover from the pandemic.

Scott Stringer, the city comptroller, said he wants to focus on the city’s affordability crisis and proposed a land bank to build affordable housing. Mr. Stringer says that the idea could help create more than 53,000 units of affordable housing on vacant land that is already publicly owned.

“Why don’t we give that land back to the people?” Mr. Stringer said at the forum.

Will the next mayor be, gasp, another Red Sox fan?

The city’s last two mayors — Mr. de Blasio and Michael R. Bloomberg — grew up in the Boston area, leading to awkward exchanges over their allegiance to the Boston Red Sox.

It seems unlikely that New York’s next mayor will hail from Massachusetts — especially after Corey Johnson, the City Council speaker, dropped out of the race — and more likely that Mr. de Blasio’s successor will be a genuine New Yorker.

Several candidates are lifelong residents, including Mr. Adams and Mr. Stringer. Kathryn Garcia, the city’s former sanitation commissioner, and Dianne Morales, a nonprofit executive, were born in the city and graduated in the 1980s from Stuyvesant High School — a prestigious public school that has received criticism for admitting only 10 Black students last year.

Mr. McGuire and Ms. Wiley grew up elsewhere, but have lived in the city for decades. Mr. McGuire was raised in Dayton, Ohio, and Ms. Wiley was born in Syracuse and raised in Washington, D.C.

Mr. Yang was born in Schenectady, N.Y., but has spent most of his adult life in Manhattan. Asked if he is a Yankees or Mets fan, he responded on Twitter last year: “Mets, unfortunately.”

The latest task for Marvel superheroes: fund-raising

The candidates are facing a major fund-raising deadline on Monday that may signal whether candidates can convert buzz into tangible support.

Ms. Wiley, in particular, is under pressure to prove that she is a top-tier candidate. She has been pushing hard for donations and received two celebrity endorsements over the weekend: the actors Chris Evans, best known for his role as Captain America in the so-called Marvel Cinematic Universe, and Rosie O’Donnell.

Mr. Stringer has another fictional superhero on his team: the actress Scarlett Johansson, who is also part of Marvel’s “Avengers” series.

Ms. Wiley has begged her supporters for donations to qualify for public matching. A candidate must raise at least $250,000 in contributions of $250 or less from at least 1,000 city residents.

Ms. Wiley wrote on Twitter that her campaign team had unleashed her to be her true self. “I can win this race and be a bad-ass Black woman mayor,” she posted.

Mr. McGuire has proved to be a prolific fund-raiser, and Mr. Stringer and Mr. Adams have already qualified for public funds. We will learn more on Friday, when campaign finance disclosure forms are released.

Is graffiti a starter crime that could have led to the Capitol attack?

As a mob of President Trump’s supporters forced their way into the Capitol on Wednesday, several of the mayoral candidates were watching and responding in real time, on social media.

Carlos Menchaca, a Brooklyn councilman, demanded the immediate arrest of the “domestic terrorists.” Ms. Wiley and Mr. McGuire argued that if the mob had been Black like them, the response from the police would have been far more violent. Ms. Morales agreed.

Then, at 3:36 p.m. that day, more than an hour after the mob had broken into the Capitol, Loree Sutton, a retired Army brigadier general and former commissioner in the de Blasio administration, struck a different note.

She highlighted an op-ed that Cardinal Timothy M. Dolan, the archbishop of New York, wrote in The New York Post, and seemed to suggest a connection between the attack on the Capitol to “vile graffiti” that recently desecrated St. Patrick’s Cathedral.

“Last month, the radicals attacked St. Patrick’s Cathedral and the N.Y.P.D.; today anarchists are attacking the U.S. Capitol … #WakeUpAmerica #StopTheMadness,” she wrote on Twitter.

In an interview, she said her words, which prompted a small uproar on Twitter, were misconstrued.

“My concern was, and it has been, that this kind of riotous vandalism, it can start in small ways, as you can point to with the graffiti, and then it can manifest and grow on either side,” Ms. Sutton said.

Nate Schweber contributed reporting from Great Neck, N.Y.