Gov. Andrew M. Cuomo was having one of the worst weeks of his administration, and he tried to divert attention to the New York City mayor’s race.

In one of his famed slide show presentations, Mr. Cuomo listed the many challenges facing the city — a rising murder rate, a homelessness crisis, people deciding to move away — and questioned if the candidates were up to the task.

“What have you managed before? What have you accomplished before?” Mr. Cuomo said in late February. “This is not about rhetoric. This is not about slogans.”

Mr. Cuomo failed to mention another top challenge for the next mayor: Figuring out how to get along with him.

Mr. Cuomo, a Democrat in his third term, has been significantly weakened by a growing crisis over allegations of sexual harassment and his handling of coronavirus-related nursing home deaths. But if Mr. Cuomo remains in office, the next mayor will have to work with him to help the city recover from the pandemic.

Now the candidates must decide how strongly to criticize Mr. Cuomo; several candidates are calling for Mr. Cuomo to face impeachment proceedings or to resign if the harassment allegations are confirmed.

But the governor is unlikely to forget those who attacked him, and Mayor Bill de Blasio’s tenure has shown how a troubled relationship with Mr. Cuomo can thwart a mayor’s agenda. Mr. de Blasio never mastered how to work with the governor, even during the darkest days of the pandemic when the leaders fought over shutting down the city.

Over the last seven years, the feud between Mr. Cuomo and Mr. de Blasio has had real implications for the city when they failed to work together on the vaccine rollout, reopening schools, a Legionnaires’ disease outbreak, and fixing the subway and public housing.

The candidates running to succeed Mr. de Blasio had been talking about how they would have a better relationship with Mr. Cuomo, but the tone drastically changed in recent days, after three women detailed accusations of sexual harassment against the governor.

On Monday, after a woman told The New York Times that Mr. Cuomo tried to kiss her at a wedding in 2019, Kathryn Garcia, the city’s former sanitation commissioner, said she had heard enough and Mr. Cuomo “should do the right thing and step aside.”

Andrew Yang, the former presidential hopeful, and Eric Adams, the Brooklyn borough president, have called for an independent investigation. Maya Wiley, a former counsel to Mr. de Blasio, criticized Mr. Cuomo for disappearing from public appearances while the state was still facing a health crisis.

“No one should throw around words like impeachment or resignation lightly,” she said. “But as a state, we must see immediate action to address the disgusting behavior” described by two female accusers of Mr. Cuomo.

Scott M. Stringer, the city comptroller, said on Saturday that he believed in a zero-tolerance policy for sexual harassment.

“I continue to support a thorough and truly independent investigation of the governor’s conduct, and if it supports these serious and credible allegations, Governor Cuomo must resign,” Mr. Stringer said.

Dianne Morales, a former nonprofit executive, and Carlos Menchaca, a City Council member in Brooklyn, have called for impeachment proceedings to begin.

The condemnations are unlikely to affect Mr. Cuomo’s involvement in the June 22 Democratic primary for mayor, which is likely to determine the winner in the general election. The governor seems unlikely to endorse a candidate; even in the presidential race, he waited until very late to endorse his friend Joseph R. Biden Jr.

Mr. Cuomo knows two leading candidates well: Mr. Stringer, a former state assemblyman, and Mr. Adams, a former state senator. Mr. Stringer has found ways to work with the governor, leading Carl Heastie, the State Assembly speaker, to call Mr. Stringer a “rubber stamp” for Mr. Cuomo in 2018 during a battle over pay raises for state lawmakers.

But Mr. Adams has openly criticized Mr. Cuomo, over his management of the subway and his feud with the mayor. In April, when Mr. Cuomo and Mr. de Blasio fought over closing the city’s schools, Mr. Adams told them to “cut the crap.” This past week, after the sexual harassment claims were lodged, he said that when powerful men prey on women, “swift action must be taken against them.”

Mr. Yang and Raymond J. McGuire, a former Wall Street executive, have said they had an in with the governor — pointing to their relationship with his brother, Chris Cuomo, the CNN host. Mr. Yang, however, voted in 2018 for Cynthia Nixon, the actress and a fierce critic of Mr. Cuomo in the Democratic primary that year — a choice that he publicized on Twitter.

Another guilt-by-association relationship could stem from one of Mr. Yang’s first endorsers: Ron Kim, the state assemblyman who has gone to war with Mr. Cuomo over virus-related deaths in nursing homes.

Having a relationship with Mr. Cuomo is not necessarily predictive of how the next mayor might get along with the governor. Mr. Cuomo and Mr. de Blasio had known each other well: Mr. Cuomo, as the secretary of the federal Department of Housing and Urban Development, hired Mr. de Blasio to oversee the New York region during the Clinton administration.

When Michael R. Bloomberg was mayor, he seemingly got along better with Mr. Cuomo. But behind the scenes, the leaders also had a tense relationship, jockeying for credit on issues like the state’s passage of a same-sex marriage law in 2011.

The stakes have never been higher. Besides dealing with the coronavirus recovery, plenty of other key issues will be decided in Albany: raising taxes on the wealthy, schools funding, fixing the subway and desegregating specialized high schools.

Democrats who have wrangled with Mr. Cuomo in the past believe that direct confrontation is the best approach. Monica Klein, a political consultant who worked for Mr. de Blasio and has organized protests outside the governor’s office, said Mr. Cuomo only responds to sustained political pressure and bad headlines.

“You can’t cede ground to a bully,” she said.

Some political consultants contend that Mr. Cuomo might prefer a candidate who has little experience in city government.

“Even a weakened Governor Cuomo would run circles around Andrew Yang,” said Eric Phillips, a former press secretary for Mr. de Blasio.

Mr. Stringer may have had an edge — at least before he brought up Mr. Cuomo’s resignation. In December, Mr. Stringer joined more than two dozen of Mr. Cuomo’s top allies in Albany, including Bill and Hillary Clinton. No other mayoral candidate was there for the historic moment: to cast the state’s official elector ballots to elect Mr. Biden as president.

Mr. de Blasio was not invited, even though the mayors of smaller cities like Buffalo and Rochester attended. Mr. de Blasio was left off the list because he ran in the Democratic primary against Mr. Biden, organizers said.

The friendly gesture did not stop Mr. Cuomo from attacking Mr. Stringer a short time later.

“The comptroller is an incumbent — where have you been?” Mr. Cuomo said to reporters in February, criticizing Mr. Stringer on police reform after huge protests in the city last summer. “What have you done? Where were you when Rome was burning?”

Mr. Stringer may also face a difficult tightrope, trying to work with the governor while answering to progressive allies who are at odds with Mr. Cuomo, including State Senators Jessica Ramos and Alessandra Biaggi — both of whom endorsed Mr. Stringer.

Mr. Stringer has made clear that he would try to be a stronger voice for the city than Mr. de Blasio.

“I’m not going to have my lunch money stolen from Albany,” he said. “You can be sure of that.”

Mr. Cuomo likely would have preferred Ruben Diaz Jr., the Bronx borough president, or Christine Quinn, the former City Council speaker, two allies who decided against running for mayor. At a recent news conference, Mr. Cuomo said he had been asked about an endorsement and wanted to know more about candidates’ plans.

“You need a real manager with a real vision who can really get things done,” Mr. Cuomo said.

Jay Jacobs, an ally of Mr. Cuomo’s and chairman of the New York State Democratic Committee, said he had given no thought to picking a favorite in the mayoral field.

Still, he added: “It doesn’t help anybody’s chances to get something from the governor, or anyone for that matter, if they’ve been spending several years banging him over the head — that’s just common sense.”

Katie Glueck and Dana Rubinstein contributed reporting.