MILWAUKEE—Donald Trump drew flak in November for billing a show of support from 100 black pastors in New York City, only to meet privately with a smaller group that included some pastors who said they weren’t supporting him.

But after that meeting, Mr. Trump gained one of his most loyal and hardworking surrogates on the campaign trail: Mark Burns, a pastor from small-town South Carolina and the owner of a fledgling Christian television network.

...