The question of whether President Trump’s endorsement power is waning in Republican primaries faces two fresh tests on Tuesday in South Carolina and Oklahoma, both solidly red states where the G. O. P. nominees will be heavily favored in the November general election. Mr. Trump’s choice to succeed the late Senator Lindsey Graham in South Carolina — Mr. Graham’s younger sister, Senator Darline Graham — is locked in a special runoff race against Representative Ralph Norman.

The president was especially fond of Mr. Graham and campaigned last week for Ms. Graham, who was appointed to the Senate last month and now seeks a full, six-year term. In Oklahoma, Mr. Trump backed Mike Mazzei, a former state lawmaker, over Gentner Drummond, the state’s attorney general, in another primary runoff, this one in the race to succeed Gov. Kevin Stitt, a Republican who is term-limited.

The value of Mr. Trump’s support has been a persistent question in the midterms as his approval rating has dropped amid voter discontent over the economy and the Iran war. His picks have yet to lose in any Senate primaries. Is Trump’s support enough to overcome Graham’s inexperience in South Carolina? Mr. Graham’s death last month forced a sprint of a special Republican Senate primary that would have been difficult for any contender.

But it was perhaps especially so for Ms. Graham. She is a first-time candidate and is little-known to G. O.

P. voters, some of whom had a complicated view of her brother over his hawkish foreign policy positions and his past support for immigration reform. Then, in the only debate leading up to the runoff, she made what was widely seen as a gaffe when she stumbled through a question about the United States’ role with Taiwan and admitted that she was “not that informed on national security.” Ms. Graham’s strongest selling point has been Mr. Trump’s endorsement, which he underscored by traveling to Myrtle Beach on Friday for a rally. “What I really want you to do is pretend, please, that I’m on the ballot,” he told Republicans. Her rival, Mr. Norman, a leader of the hard-line House Freedom Caucus whose main focus is spending cuts, had spent the past year campaigning for governor.

He finished third in that race in June and then turned his attention to the Senate. Outspent by Ms. Graham and a super PAC that supports her, he has been careful not to pointedly attack her or Mr. Trump, focusing instead on his experience as a five-term congressman. The campaigns will be taking a close look at which Republicans make it to the polls.

Only the most partisan and informed of voters tend to vote in primary runoffs, particularly in special elections, and particularly in August. Mr. Norman’s strength is expected to be in South Carolina’s Upstate region, which includes Greenville and Spartanburg. Ms. Graham is likely to do better elsewhere. Of note for results watchers: Greenville County, which is rich in Republican voters, tends to report slightly later than other counties in the state. And what might it mean for Trump if it’s not? Not only did Mr. Trump explicitly ask Republicans to think of voting for Ms. Graham as if he were on the ballot, but a super PAC run by his advisers spent nearly a third of its total outlay so far this year just on this race.

The super PAC, MAGA Inc., disclosed on Monday that it had spent nearly $800,000 on Ms. Graham’s behalf, highlighting the importance of Ms. Graham’s candidacy to the president. Were Ms. Graham to lose, Mr. Trump’s supporters will surely note the unusual nature of the contest — a special Senate primary run in an extraordinarily short period of time — and suggest the loss means little for Mr. Trump’s popularity or his midterms strategy more broadly. Others are likely to speculate that the president’s hold on his party is slipping alongside his approval numbers.

There’s also the question of what it means for Mr. Trump’s agenda to lose an ally in the Senate. Mr. Norman has staunchly backed Mr. Trump’s agenda over ...