She has expressed support for Mr. Trump in the past, though not as fervently as some of the Republicans she bested on Tuesday. In a brief victory speech, she did not mention Mr. Trump at all.

But in an appearance Wednesday morning on CNN, Ms. Handel said she hoped Mr. Trump would come campaign for her. “It’s all hands on deck now,” she said. “There’s a lot at stake here.”

Mr. Trump is expected to be in the Atlanta area on April 28 to speak at a National Rifle Association event. It appears, from his Twitter account, that he has taken a keen interest in the race, and has offered his assistance: “Despite major outside money, FAKE media support and eleven Republican candidates, BIG ‘R’ win with runoff in Georgia,” he wrote Tuesday. “Glad to be of help!”

But even if he does not stump for Ms. Handel, his presence will be felt keenly in the runoff, with both candidates facing unique Trump-related dilemmas over the next few weeks.

If Ms. Handel keeps a cool distance from the president, she risks alienating the most fervent pro-Trump voters. If she embraces him, she may lend ammunition to Democrats eager to portray Ms. Handel as his tool.

Mr. Ossoff’s problem has to do with outrage. The anger over the Trump presidency has been the rocket fuel that has powered his surprisingly successful campaign, unleashing not only a flood of out-of-state donations, but a formidable army of canvassers and telephone-bank volunteers.

Whether that level of passion and involvement is sustainable remains to be seen. Democrats have a poor track record of turning up at midterm and special elections, and the June runoff comes during peak vacation season.