A primary challenge to the speaker of the Texas state House, Dade Phelan, that is being backed by the state’s attorney general, Ken Paxton, is set to be the most expensive state House race ever, with outside groups pouring money into the effort to unseat the incumbent.

Ever since Mr. Paxton’s impeachment in the Texas legislature, which saw the attorney general acquitted of corruption and bribery charges, Mr. Paxton has been on a mission to oust Republican state lawmakers who backed the effort to remove him.