President Trump’s endorsed candidates won Republican primary runoffs for Senate on Tuesday in Georgia and Alabama, but his choice for governor in Georgia fell to defeat.

In Georgia, Representative Mike Collins, whom Mr. Trump backed only on Sunday morning, defeated Derek Dooley for the Senate nomination. Lt. Gov. Burt Jones, a longtime Trump ally whom the president endorsed last August, lost to Rick Jackson, a self-funding businessman. In Alabama, Representative Barry Moore turned back Jared Hudson, a retired Navy SEAL.

Mr. Collins will face Senator Jon Ossoff, the lone Senate Democrat running for re-election in a state Mr. Trump won in 2024, in the general election. Mr. Jackson’s Democratic opponent is Keisha Lance Bottoms, the former Atlanta mayor.

Mr. Moore, who has a far easier road to general-election victory in deep-red Alabama, is likely to replace Senator Tommy Tuberville, who won the state’s Republican nomination for governor last month.

In an interview late Tuesday, Mr. Moore credited his victory to Mr. Trump’s influence, recounting that many Alabamians had told him, “If the president endorsed you, we’re with you.”

Here’s what to know about Tuesday’s other races.

• Oklahoma Senate: Representative Kevin Hern of Oklahoma won his Republican primary for an open Senate seat. Mr. Trump helped clear the field by endorsing Mr. Hern after he elevated Markwayne Mullin to be his homeland security secretary.

• Oklahoma governor: Oklahoma’s Republican primary for governor is headed for a runoff in August. Gentner Drummond, the state attorney general, and Mike Mazzei, the Trump-endorsed former state legislator, placed first and second in the nine-way race.

• Oklahoma First Congressional District: Another Oklahoma primary with a Trump endorsee will head to a runoff: Jackson Lahmeyer, who started the group Pastors for Trump, placed second behind Mark Tedford, a state legislator. The winner is likely to replace Mr. Hern.

• Washington, D.C., mayor: Late Tuesday, votes were still arriving in the primary contests for an open-seat race for mayor. The leading Democrats to succeed Mayor Muriel Bowser, who is not seeking re-election, are Janeese Lewis George, a progressive council member, and Kenyan McDuffie, a relatively moderate former City Council member.