Eric Barlow’s defeat of Megan Degenfelder, the state superintendent of education, again showed the limits of the president’s favor.

Ms. Hageman, who unseated Representative Liz Cheney in 2022, is a heavy favorite to replace Senator Cynthia Lummis in the deep-red state.

© Valerie Plesch for The New York Times

Four states are holding elections on Tuesday. Election officials anticipate having results by late Tuesday or early Wednesday.

© Scott McIntyre for The New York Times