The president appears to exist in a sunny political environment all his own. In a late-night tweet on Tuesday, Mr. Trump bragged about “a big turn for the better” in Mr. Balderson’s campaign after the president held a boisterous get-out-the-vote rally in Ohio on Saturday night. On Wednesday morning, he crowed that Republicans went “5 for 5” in races on Tuesday and predicted that whenever he campaigns for candidates, “they will win!”

“As long as I campaign and/or support Senate and House candidates (within reason), they will win! I LOVE the people, & they certainly seem to like the job I’m doing. If I find the time, in between China, Iran, the Economy and much more, which I must, we will have a giant Red Wave!” he boasted Wednesday on Twitter.

It is impossible to know for sure the impact of Mr. Trump’s last-minute help in an Ohio district that Republicans should have won easily. The president could be right that he lifted turnout among Mr. Balderson’s supporters just enough to secure the win, if and when it is declared. Or it’s possible that his rally did nothing much — polls ahead of Election Day suggested a one-point race, which is about where it ended up.

Nor is it possible to gauge the impact of a late-tweeted endorsement of Mr. Kobach, the divisive Kansas secretary of state who headed the president’s voter integrity commission before challenging the sitting Republican governor.

The president’s political advisers said internal polling in Mr. Balderson’s race showed him trailing, and turned around only after Vice President Mike Pence visited and the president held the last-minute rally. They noted that far more Democrats were voting early than Republicans in Ohio until those White House visits.