Mr. Trump has also backed candidates in crowded open-seat primaries for governor in Georgia and the Senate in Alabama that are expected to end up in runoff elections if no candidate wins more than 50 percent of the vote. The president has notably not weighed in on Republicans’ important primary for the Senate in Georgia, where the party is aiming to unseat Senator Jon Ossoff, a Democrat.

Idaho, Oregon and Pennsylvania will also hold primaries, with a Democratic contest for one Pennsylvania House district attracting the most attention. Democrats are also choosing their nominee for governor in Georgia, where former President Joseph R. Biden Jr. has weighed in to endorse a candidate in a race that may also go to a runoff.

Here are eight questions about Tuesday’s action:

Can President Trump purge a Republican antagonist in Kentucky?

Known as “Mr. No” in Washington for his years of rebelling against House Republican leadership, Mr. Massie earned Mr. Trump’s enmity when he opposed his major domestic policy bill, campaigned for the Democratic-led effort to release the Epstein files and became a vocal opponent of the war with Iran.

Mr. Trump endorsed a challenger, Ed Gallrein, a retired Navy SEAL, and traveled to Kentucky to hold a rally for him.

Days after Senator Bill Cassidy of Louisiana lost his Republican primary because he crossed Mr. Trump, the Kentucky race will offer new evidence of just how tightly the president still controls his party. Mr. Massie has not shied away from the challenge, emphasizing his independence.

Mr. Trump’s allies have poured millions into the race, making it the most expensive House primary in recent years with more than $30 million spent on advertising, according to data from AdImpact, a media tracking firm.

Unlike in Indiana, where Mr. Trump pushed out a series of Republican state legislators after they resisted his redistricting drive, Mr. Massie has inspired robust spending on his behalf. It has given him an opportunity, if he manages to win, to become the leader of a Trump opposition movement within the Republican Party.

Can a Trump endorsement outweigh a billionaire’s cash in Georgia?

Usually, a Trump endorsement in an open-seat race will clear a Republican primary field. But in Georgia, a wealthy health care executive is testing how much the president’s backing is worth.

Mr. Trump endorsed Lt. Gov. Burt Jones in the Republican primary to succeed Gov. Brian Kemp, a Republican who cannot run again because of term limits.

At the time, Mr. Jones’s main Republican rival appeared to be Brad Raffensperger, the Georgia secretary of state, who resisted Mr. Trump’s efforts to overturn the result of the 2020 presidential election in the state.

But then Rick Jackson, a billionaire former business partner of Jeb Bush, entered the race in February. He has since spent $83 million of his own money, much of it on TV ads slamming Mr. Jones as a career politician.

Mr. Jackson and Mr. Jones have led in most polling, but with Mr. Raffensperger and other candidates in the race, it is likely to head to a June 16 runoff.

And what is a Biden endorsement worth in 2026?

Former President Joseph R. Biden Jr. has largely been quiet since leaving office.

But he did emerge this month to endorse Keisha Lance Bottoms, a former Atlanta mayor and Biden administration official who is in a crowded battle for the Democratic nomination for Georgia governor.

Compared with the gushers of cash from Mr. Jackson and Georgia Republicans, the Democratic candidates have spent a pittance on their campaigns. Polling suggests that Ms. Lance Bottoms is the favorite to place first in the primary, and that she has a shot at winning a majority and avoiding a runoff.

Her competition includes former Lt. Gov. Geoff Duncan, a former Republican who broke with Mr. Trump and switched parties to run for governor; Jason Esteves, a former state senator; Derrick Jackson, a state representative; and Michael Thurmond, who was the party’s nominee for the Senate in 2010.

What about a Brian Kemp endorsement?

With two congressional allies running, Mr. Trump has stayed out of Georgia’s Republican primary for the Senate, in which the winner will face Senator Jon Ossoff, a first-term Democrat.

But Mr. Kemp has waded in, spending his political capital and activating his campaign apparatus to help Derek Dooley.

Mr. Dooley, a former head football coach for the University of Tennessee whose father was football coach and athletic director for decades at the University of Georgia, is now in a dogfight (dawgfight?) with Representatives Mike Collins and Buddy Carter. With two other candidates also running, this race is also likely to head to a June runoff.

Polls suggest Mr. Collins is the most likely of the three to advance to a runoff.

Can Trump pick senators in Alabama and Kentucky?

Tommy Tuberville, a former college football coach turned Alabama senator, is expected to cruise to the Republican nomination for governor. Former Senator Doug Jones is the heavy favorite to prevail in the Democratic primary for governor.

The race for Mr. Tuberville’s seat in the Senate is more complicated.

Mr. Trump endorsed Representative Barry Moore. But as in Georgia’s governor’s race, the Trump endorsement has not scared rivals away.

Steve Marshall, the Alabama attorney general, and Jared Hudson, a retired Navy SEAL, are among the Republicans running for the seat. Mr. Moore has spent the most money, but not by much, and the race could head to a runoff.

Whoever wins the Republican primary will be a heavy favorite in deep-red Alabama.

Mr. Trump is facing less resistance in Kentucky, where he endorsed Representative Andy Barr to succeed Senator Mitch McConnell, who is not seeking re-election. Mr. Barr’s top rival, Nate Morris, dropped out of the race this month after the Trump endorsement. Daniel Cameron, a former Kentucky attorney general and longtime McConnell ally, is still in the race.

What happens when Josh Shapiro and Bernie Sanders endorse the same candidate?

The most interesting Democratic primary on Tuesday is in the Lehigh Valley in Pennsylvania, where four Democrats are competing to challenge Representative Ryan Mackenzie, a Republican who won election in 2024 by just 4,000 votes.

Bob Brooks, a firefighters union president, has an unusual coalition of supporters, including Gov. Josh Shapiro of Pennsylvania, Senator Bernie Sanders of Vermont and Pete Buttigieg, the former transportation secretary. House Democrats’ campaign arm has also endorsed Mr. Brooks.

In ordinary times, that combination of establishment and left-wing support would suggest that Mr. Brooks was certain to win the primary. But he has real challengers, including Ryan Crosswell, a former Justice Department lawyer who has backing from VoteVets, a liberal veterans group, and Lamont McClure, a former Northampton County executive with less funding, who in recent weeks has been the beneficiary of ads from a Republican super PAC trying to lift his chances in the primary.

Will court candidates backed by Barack Obama and Kamala Harris win?

If Democrats needed any reminding of the importance of state supreme courts, they got it this month when Virginia’s high court threw out a congressional map voters had recently approved in a referendum.

Now, amid the primary elections, two Republican-appointed justices on the Georgia Supreme Court are up for re-election on Tuesday, with challengers backed by former President Barack Obama, former Vice President Kamala Harris and other leading Democrats hoping to tilt the balance of the formally nonpartisan court.

Eight of the nine current Georgia Supreme Court justices were appointed by either former Gov. Nathan Deal, a Republican, or Mr. Kemp, the current Republican governor.

Will Oregon Republicans take their cue from the co-founder of Nike?

Deep-blue Oregon and deep-red Idaho are also holding primaries on Tuesday.

In Oregon, Gov. Tina Kotek, a Democrat, is running for re-election but will first need to defeat several primary challengers. A crowded field is vying for the Republican nomination, including Christine Drazan, a former state legislator who lost to Ms. Kotek four years ago, and Chris Dudley, a former professional basketball player who was the party’s nominee for governor in 2010.

Mr. Dudley is backed financially by Phil Knight, the billionaire co-founder of Nike, the shoe company.

In Idaho, Gov. Brad Little, who is endorsed by Mr. Trump, seems likely to cruise through his primary on his way to a third term.