Mr. Paterson, mired in constant controversy, had become an unwanted distraction for the Obama administration; he soon announced he would not run again, opening the way for Mr. Cuomo to run in 2010.

But there was a significant difference between Mr. Paterson and Mr. Cuomo: Mr. Paterson’s poll numbers were dismal, at 21 percent approval in June 2009, and threatened the Democrats’ hold on a seat that a Republican, George Pataki, had held for three terms, until 2006.

Mr. Cuomo has much stronger support from his constituents. A new Siena College poll on Monday showed that only 35 percent of New York voters want Mr. Cuomo to resign immediately (and only 25 percent of Democrats), though the survey was taken mostly before the wave of congressional demands for him to quit.

Still, support for Mr. Cuomo has eroded significantly from the highs of his coronavirus press briefings in the spring of 2020 — when he hit 71 percent approval — and even from February, when his approval among all voters was 56 percent in a Siena College poll.

His current approval rating, 43 percent, is lower than his disapproval rating, 45 percent. Among Democrats, however, his support remains high, at 59 percent, the survey showed, and 61 percent among Black voters.

Already, a majority of state legislators — and more than 40 percent of Democratic lawmakers in Albany — have called for Mr. Cuomo’s resignation. The State Assembly has launched an impeachment inquiry, and beyond Mr. Biden, the politician with the most control over Mr. Cuomo’s fate is the Assembly speaker, Carl E. Heastie, who will determine if and when to proceed.

Impeaching and removing a governor is a serious undertaking, and Mr. Cuomo can hope that it is too big a leap even for those who signed a letter urging his resignation.

“Calling the Legislature’s bluff on an impeachment vote, he recognizes casting a vote for impeachment is a heavy vote for many to make,” said Bakari Sellers, a former member of the South Carolina House of Representatives, who voted to impeach then-Gov. Mark Sanford in 2009. (Mr. Sanford was eventually censured.)

“The state is about to be flush with Covid cash,” Mr. Sellers said. “Better days ahead for constituents. Hang on until you become everyone’s favorite bank.”

Some members of Congress and their aides were put off by a statement that the former senior member of the delegation, Representative Nita Lowey, of Westchester, had made in defense of Mr. Cuomo, according to a person familiar with the matter. The members had the sense that Ms. Lowey was inappropriately “instrumentalized” as a shield for Mr. Cuomo, the person said, adding that while it wasn’t the precipitating event for other members to speak out last Friday, it had left an impression.

A member of the Cuomo family had reached out to Ms. Lowey before her statement, according to another person familiar with the events.

“That’s ridiculous,” Ms. Lowey. “I don’t get used for things like that.” She said she’s known the Cuomos for decades, since they were neighbors in Queens.

Mr. Biden and Mr. Cuomo have been relatively close politically in recent years. In 2015, when Mr. Biden was considering a belated run for president, they met in New York; though Mr. Cuomo was formally backing Hillary Clinton at the time, he did not discourage Mr. Biden from a White House run.

In 2018, when Mr. Cuomo faced a primary challenge of his own from Cynthia Nixon, the actress and activist, Mr. Biden offered a full-throated endorsement of Mr. Cuomo at the New York Democratic Party convention.

Mr. Biden’s fondness for Mr. Cuomo does not necessarily extend to the staff level. The governor’s sharp-elbowed political operation has jabbed at many people in his path over the years.

Mr. Biden tapped Mr. Cuomo for a prime-time speech on the first night of the Democratic convention last year at the height of the governor’s popularity. The prerecorded address, which did not mention Mr. Biden by name until near the end, was thrust on convention organizers with little opportunity for revision, according to people involved in the process; they said the Cuomo team was among the most difficult to work with in planning the entire four-day event.

Mr. Cuomo’s political operation also submitted a production bill that far exceeded other similar convention videos; convention officials declined to pay the full amount.

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