GREEN, Ohio—Ohio Republicans are poised Tuesday to decide whether to hand Donald Trump a commanding advantage in the race for the presidential nomination or help their governor create a multicandidate scramble for the party’s remaining convention delegates.

Gov. John Kasich, locked in what public polling shows to be a close race with Mr. Trump, stumped Monday with the GOP’s last nominee, Mitt Romney, who offered a near-endorsement at his first presidential campaign appearance since 2012.

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