The big political story in New York City is the growing crisis surrounding Gov. Andrew M. Cuomo, who faced new allegations of sexual harassment over the weekend.

Several Democratic mayoral candidates responded with calls for an independent investigation, and some said Mr. Cuomo, a Democrat, should resign if the allegations are substantiated.

And as the Republican field begins to take shape, and many candidates are holding more in-person events, the contest — with the primaries now just four months away — is starting to feel a lot like a normal election, even with the coronavirus still a concern.

Here’s what you need to know about the race:

A rebuke for Cuomo

Many candidates responded to a New York Times article disclosing that a second woman had accused Mr. Cuomo of sexual harassment by calling for an independent investigation, expressing disgust and demanding his resignation if the allegations were further substantiated.

Charlotte Bennett, 25, a former executive assistant and health policy adviser for the governor, said he asked questions about her sex life, including whether she had ever had sex with older men. The charges come after Lindsey Boylan, a former state economic development official, accused Mr. Cuomo of giving her an unwanted kiss.

Mr. Cuomo called Ms. Boylan’s allegations untrue and said he was sorry that some of the things he had said to Ms. Bennett “have been misinterpreted as an unwanted flirtation.” The governor is also facing questions over how he handled the state’s nursing homes during the pandemic and over charges of bullying behavior.

Among the candidates, Scott M. Stringer and Raymond J. McGuire went the furthest, calling for Mr. Cuomo to resign if an independent investigation substantiated the sexual harassment allegations.

Mr. McGuire called the allegations “deeply disturbing” and said the accused conduct was “abhorrent.” He said the governor “should resign” if they were further substantiated.

Mr. Stringer said the governor “must resign” if an investigation “supports these serious and credible allegations.”

Dianne Morales had already called for impeachment proceedings to begin against Mr. Cuomo because of allegations of bullying and the way he had handled nursing homes.

“It’s time to address the complete abuses of power that Cuomo has exercised for far too long,” Ms. Morales said in a statement.

Kathryn Garcia, Andrew Yang and Shaun Donovan called for independent investigations into the allegations.

Mr. Yang said that victims of sexual harassment should “feel empowered” to share their stories “without fear or retaliation” and that “Albany must show they take all allegations seriously through action.”

Ms. Wiley registered her disgust in a statement on Saturday, saying, “I believe Charlotte Bennett.” She followed up with a statement on Sunday that had at least 20 or so questions about the situation.

Republicans jockey for endorsements

Another Republican has entered the mayoral race: Curtis Sliwa, the red beret-wearing founder of the Guardian Angels, who is running on a law-and-order message.

Mr. Sliwa registered with the city Campaign Finance Board recently and was endorsed by Republican leaders on Staten Island. That prompted Fernando Mateo to announce endorsements from the Republican Party in Manhattan, the Bronx and Queens. The two are expected to be top contenders for the Republican nomination, though either would be an extreme long shot in the general election, given that the vast majority of voters in the city are Democrats.

“The reason I’m running for mayor is our city is a ghost town,” Mr. Sliwa said in an interview on Newsmax, criticizing rising crime and homelessness.

Mr. Sliwa knocked Mr. Yang, the former presidential candidate, saying he wants to “give out money that we don’t have” — a reference to universal basic income — and he argued that the Democratic field wants to defund the police. (Most of the Democrats have been reluctant to embrace the defund movement, but they do want to change department policies.)

Mr. Sliwa said the Police Department had been “neutered” since George Floyd’s death in Minneapolis last year, and he promised to restore police funding and boost morale by visiting every precinct as mayor.

“I’ll pat these cops on the back so hard they’ll have to go for a chiropractic adjustment,” he said.

Mr. Mateo, who was born in the Dominican Republic, has highlighted his appeal to Hispanic voters. The Bronx Republican Party said in a statement that Republicans had made “significant inroads” in minority communities, especially with Hispanics.

“With Mateo at the top of the Republican ticket in 2021, we can replicate that success citywide and continue to expand the Republican coalition,” the group said.

Ditching the video campaign, if only for an afternoon

There are walking tours and outdoor lunches, policy rollouts and church visits.

As the weather begins to warm and the primary election nears, the Democratic mayoral candidates are slowly getting back out onto the campaign trail, appearing increasingly willing to balance the risks of campaigning in a pandemic with the need to engage and excite more voters beyond Zoom.

In the last week, all of the leading mayoral candidates made campaign stops across the city, in some cases several stops in one day. Ms. Wiley spent Friday afternoon in the Bronx; Mr. Stringer offered his housing plan outdoors; Mr. McGuire spent Saturday campaigning in southeast Queens.

Mr. Yang, who had to quarantine after testing positive for the coronavirus last month, has from the beginning of his campaign shown more comfort with in-person events, and he was the pacesetter again last week, taking a five-day tour through the five boroughs.

The strategy can pay off. Mr. Yang rode the Staten Island Ferry and made positive headlines for helping defend a photographer from an attack.

Eric Adams, the Brooklyn borough president, and Carlos Menchaca, a Brooklyn city councilman, even held an unusual joint campaign event in front of the Phoenix Hotel in Sunset Park, Brooklyn, to call for converting empty hotels into affordable housing.

Mr. Adams said in an interview that his long days usually start with meditation at 6 a.m., and a recent evening ended with a dinner with South Asian leaders at 9 p.m.

“Every second is utilized,” he said. “From the time I wake up to the time I hit my pillow.”

A campaign of (stolen) ideas

One candidate accusing another candidate of appropriating their campaign’s ideas: It’s a time-honored complaint on the trail. The Democratic primary for mayor is no different. Several candidates for mayor are proponents of some version of universal basic income, one of Mr. Yang’s campaign platforms from his run for president. Now, some candidates are accusing Mr. Yang of pilfering their campaign ideas.

Mr. Adams’s campaign faults Mr. Yang’s campaign for stealing ideas to provide pregnant women with doulas and use shuttered storefronts as vaccine distribution centers.

Stu Loeser, an adviser on Mr. McGuire’s campaign, accused the Yang campaign of appropriating their idea to allow small businesses to keep their sales tax receipts for a year to help recover from the pandemic, and to create a teachers’ corps to tutor students.

Mr. McGuire’s campaign grew so annoyed that they decided to do a bit of internet trolling: Anyone who heads to yangpolicy.com is automatically redirected to Mr. McGuire’s campaign website. The official registration for the web address is anonymous, but Mr. McGuire’s campaign claimed credit.

“Lots of candidates say they will take on wasteful duplication. We set up yangpolicy.com to actually do something about it,” said Lupe Todd-Medina, a spokeswoman for Mr. McGuire.

Mr. Yang’s campaign ridiculed the accusations, suggesting that Mr. McGuire’s campaign and others have used ideas they first proposed.

“You know what’s not a new idea? Last-place candidate going after first-place candidate to get attention,” said Alyssa Cass, Mr. Yang’s communications director.

As for Mr. Adams’s idea about doulas, Ms. Cass said Mr. Yang agreed with Mr. Adams and had spoken with others about the idea.

“Eric has had 15-plus years as an elected official and never gotten it done,” Ms. Cass said. “We’ll make it a Year 1 priority.”

Who will save Broadway?

One central issue in the race is how to bring back Broadway and the city’s struggling cultural institutions.

Ms. Garcia, the former sanitation commissioner, released her plan last week called “Reopen to Stay Open,” which calls for removing red tape for small businesses and working with streaming services to broadcast Broadway shows.

Ms. Garcia said she will be the city’s cheerleader, visiting museums and Broadway shows to get New Yorkers excited about returning to them. She held a recent Broadway-themed fund-raiser with Will Roland, an actor from the musical “Dear Evan Hansen.”

Ms. Wiley, a former counsel to Mayor Bill de Blasio, has talked about rebuilding the arts and proposed spending $1 billion on a recovery plan for artists and culture workers as part of her “New Deal New York” proposal.

Ms. Wiley also said she wants to be a cheerleader for the city and would not run away from the job, referring to Mr. de Blasio’s penchant for spending time outside the city during a failed presidential run in 2019.

“You don’t have to worry about me going to Iowa,” Ms. Wiley said at a candidate forum. “I’d much rather be on Broadway celebrating its survival.”