HENNIKER, N.H.—John Kasich, killing time between media interviews on the campaign trail Wednesday, threw a bad game of darts. But once he focused, he threw two bull’s-eyes in a row.

It serves as a metaphor for how he has been running for the GOP presidential nomination. One of the last candidates to enter the race, Mr. Kasich has been hyper-focused on New Hampshire, and it is starting to pay off.

The...