AUSTIN, Texas — With President Trump arriving in red-state Texas for a campaign rally in Dallas on Thursday, the Republican Party in the state faces a host of troubles.

The Republican-controlled Texas House of Representatives is engulfed in scandal. Six of the state’s 23 Republican members of the United States House of Representatives say they will not run for re-election, opening new opportunities for Democrats. And one of the state’s three top Republican leaders believes that the president has become a political liability among a crucial bloc of voters.

“With all due respect to Trump — who I love, by the way — he’s killing us in urban-suburban districts,” Dennis Bonnen, the speaker of the state House and the central figure in the legislative scandal, said in a 64-minute tape recording released on Tuesday.

The recording of a sometimes salty conversation Mr. Bonnen had with a conservative activist at the State Capitol in June includes a description of what critics have called a quid-pro-quo offer that is now under investigation by the Texas Rangers.