The vice president was about to go onstage and deliver a campaign speech in Michigan when her aides got an unexpected heads up: The president was about to address reporters in Washington.

President Biden entered the White House briefing room for the first time in his term last Friday, promoting a better-than-expected jobs report just as Vice President Kamala Harris was to address union workers and other supporters in Flint.

As he detailed the numbers, he extended credit to Ms. Harris. But he could not resist taking a question about whether he would reconsider dropping out of the presidential race.

“I’m back in,” Mr. Biden said, laughing before cracking a smile and offering a wave of his hand, as if to say, “I wish.”

Within Ms. Harris’s campaign, reactions to the president’s impromptu appearance ranged from calling it unhelpful to genuine anger and incredulity, according to several aides who insisted on anonymity to detail private reactions.

Ms. Harris has under a month left to make her case to the American people that she should lead the nation. But even as she tries to show that she is more than Mr. Biden’s understudy, moments like these are a reminder of the awkward dance steps required of Ms. Harris to both serve a boss and seek to replace him.