Though the Cuban vote has been historically Republican, the political allegiances of younger Cubans have been shifting, making the group far less reliable for Republican candidates than it used to be. According to the Pew Research Center, less than half of Cuban-American registered voters nationwide — 47 percent — say they identify with or lean toward the Republican Party. A decade ago, nearly two-thirds of them did.

Mr. Trump’s allies in Florida have dismissed the Newsweek report. His campaign chairman in Miami-Dade County, Lorenzo Palomares, told the Spanish-language newspaper Diario Las Americas that the article was a political ad “paid for by Hillary Clinton.”

But the Clinton campaign has been relentless in trying to keep the story alive. After Mr. Trump’s campaign manager, Kellyanne Conway, appeared on “The View” and seemed not to know much of the details of the $68,000 — “I think they paid money, as I understand from the story, they paid money in 1998,” she said — the Clinton campaign took that as an admission of guilt.

Jake Sullivan, Mrs. Clinton’s senior policy adviser, issued a statement that accused Mr. Trump of lying about his real intentions in Cuba.