Here are the other Texas runoff elections we’re watching.

Texans voted on Tuesday in runoff elections for a number of offices, with competitive primary races for two congressional seats and a tough fight playing out among Republicans for the seat held by the speaker of the State House.

All of Tuesday’s contests happened because no candidate received a majority of the vote in the primaries in March.

Jay Furman, a retired Navy officer who served in the military for nearly 30 years, prevailed over Lazaro Garza Jr., a rancher, in the Republican primary to challenge Representative Henry Cuellar, a centrist Democrat who has been indicted on bribery charges.

In a separate race on Tuesday, Mr. Cuellar’s sister — Rosie Cuellar, a former county tax assessor — is running in a Democratic primary for a state House seat. That contest could offer some hints at how much the Cuellar name has been tarnished by his indictment.

Here are two other races we’re keeping on tabs on:

Representative Tony Gonzales, a Republican who recently called some far-right members of his party “scumbags,” was forced into a runoff against Brandon Herrera, a gun rights activist, after receiving only 45 percent of the primary vote. Mr. Herrera received about 25 percent in a five-person race.

Mr. Gonzales was first elected in 2020 to represent a swing district along the border, but the seat was made more solidly Republican through redistricting. And Mr. Gonzales has bucked the party line on some issues: He voted for bipartisan gun control legislation after the massacre at a school in Uvalde, Texas, and for a gay marriage bill. He also once opposed hard-line immigration policies, and the Texas Republican Party censured him — but he has since shifted toward them.

Mr. Herrera, a YouTuber who calls himself “the AK guy” after the assault rifle, is hoping to consolidate the votes that went to multiple right-wing candidates who opposed Mr. Gonzales in the first round. He has been endorsed by Representative Matt Gaetz of Florida and other members of the House Freedom Caucus, but House Republican leaders are backing Mr. Gonzales.

The speaker of the Texas House, Dade Phelan, is in a tight Republican contest with David Covey, a former county party leader who has never run for state office before. Mr. Covey is backed by several wealthy donors and has been endorsed by Mr. Trump, whose blessing has played a large role in Republican primaries.

Mr. Covey narrowly led Mr. Phelan, 46 percent to 43 percent, in the first round of voting.

The opposition to Mr. Phelan is driven in large part by the state attorney general, Ken Paxton, who is seeking revenge for the Texas House’s vote to impeach him on charges of corruption and abuse of office. Mr. Paxton was acquitted by the Texas Senate and has campaigned for Mr. Covey. Others see an opportunity to shift the Legislature even further to the right.