Younger black voters, in particular, have expressed misgivings about Mrs. Clinton because of some of the policies of her husband’s administration.

These voters specifically point to the 1994 crime bill, which put more police officers on the streets, but also led to tougher sentences for nonviolent drug offenders and the overhaul of welfare, which reduced federal assistance for the poor by nearly $55 billion over six years.

In addition, it is hard for Mrs. Clinton to replicate the deep personal affection and pride that many African-Americans feel for Mr. Obama.

“People say, ‘It doesn’t matter because Hillary Clinton will get 90 percent of the African-American vote,’” said Charlie King, a prominent New York Democrat. “The question is, ‘Ninety percent of what?’ Turnout makes a difference.”

With 50 days until the election, Minyon Moore, a senior adviser to the Clinton campaign, called Mr. Obama’s remarks on Saturday “a true wake-up call” to his coalition of supporters.

In part, the speech reflected the president’s eagerness to use the outsize sway he has with black voters, especially younger people who had not been engaged in politics before his bids for the White House.

But in recent days, advisers to Mr. Obama say, the president has grown exasperated with the tenor of the campaign — including the re-emergence of questions about his birthplace, an issue that he and many of his supporters have long regarded as racist. While his advisers have suggested his most powerful role in the Clinton campaign is as a convert who has come to respect his onetime rival, Mr. Obama is also determined to warn his supporters about the dangers of failing to turn out and essentially ceding the presidency to Mr. Trump.