In any election, different types of voters are likely to be swayed by different factors. Liberals, for instance, often express more concern about health care, whereas conservatives generally pay more attention to foreign affairs. Among ambivalent swing voters, it is often candidates’ personalities — whether they are perceived as relatable, trustworthy, evenhanded and so on — that make a difference.

But this year’s presidential race may be an anomaly. With the response to the pandemic taking center stage, issues like health care, the economy and voting rights are likely to be viewed through the lens of the virus. So how the president handles the response — and how his presumptive opponent, former Vice President Joseph R. Biden Jr., proposes to confront the outbreak — could become a kind of omni-issue.

At his daily news conferences, and in frequent tweets about the virus, Mr. Trump has sought to project authority as the nation weathers the storm. But he has hardly let up on his habit of lashing out at political opponents, and he has refused to take responsibility for things like shortages of test kits and medical supplies, which many governors say they need.

Most Americans are not convinced of the president’s position. Roughly seven in 10 said the federal government should be doing more to address shortages in personal protective equipment and medical devices, the CNN poll found. And by a 14-percentage-point margin, most respondents to that survey said the federal government had done a bad job of preventing the virus’s spread. (That is a significant change since last month, when back-to-back CNN polls found the public more split on the question.)

As the pandemic has worsened nationwide, Americans have expressed widespread approval of their own governors (72 percent, according to a Monmouth University poll released this week) and of federal health agencies (66 percent, per the Monmouth poll). This bucks the trend in recent years of waning faith in American institutions.

But Mr. Trump has never shed his public persona as an anti-establishment firebrand. Throughout the crisis he has criticized everyone from nonpartisan inspectors general to popular state governors. So the president will need to contend with Americans’ belief in core institutions — which Mr. Biden has consistently echoed — alongside their wavering confidence in Mr. Trump himself.

“He likes to say, ‘Well, the governors messed up,’ and ‘China lied to me,’ and all this other stuff,” Bob Shrum, the director of the Center for the Political Future at the University of Southern California, said in an interview. “I think it’s a little like Herbert Hoover saying, ‘The Great Depression, you know, that’s really a world problem.’ People’s reaction was: ‘Well, wait. You’re the president of the United States. Fix it.’”