“The New York primary and the psychological impact it will have on the rest of the country and the rest of this race is not — I don’t have the words to explain it,” Mr. Clinton, who is not typically at a loss for words, told members of the Building and Construction Trades Council of Greater New York on Thursday afternoon in the city’s Flatiron district. “I can just tell you that, for her, it means more than you will ever know.”

As Mr. Sanders hopes for an upset to humiliate Mrs. Clinton on her home turf and revive his underdog candidacy, the Clintons are grinding for a New York victory that will solidify her delegate lead and put a proverbial lid on any doubts about her candidacy.

“Look, this primary is really important,” Mr. Clinton said Thursday. “It’s important to her personally because she loved being a senator, and she knows this state and she knows what its promise is.”

By the late afternoon, Mr. Clinton was talking to teachers in Battery Park City and reciting exact polling data from when Mrs. Clinton ran for the Senate in New York in 2000 (“The last poll said she was three points ahead,” he said. “She won by 12.”) and about her approval rating (“When she left the State Department job, she had 69 percent approval rating.”)

Mr. Clinton’s work on behalf of his wife in New York has also been waged quietly and behind the scenes as he taps into decades-long relationships to help her win in the state.