WASHINGTON—President Trump is signaling that a focus on racial and cultural divides will be a key theme of his re-election campaign. But with the nation in the midst of a racial justice movement and polls showing a growing portion of the population supportive of change, some Republicans fear his strategy won’t work.

Over the past few weeks, Mr. Trump questioned a Nascar ban on the Confederate flag and mounted a defense of military base names honoring Confederate officers. He criticized the Black Lives Matter movement and retweeted…

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