In early 2015, a man who runs a small technology company showed up at Trump Tower to collect $50,000 for having helped Michael Cohen, then Donald Trump’s personal lawyer, try to rig online polls in his boss’s favor before the presidential campaign.

In his Trump Organization office, Mr. Cohen surprised the man, John Gauger, by giving him both a blue Walmart bag containing between $12,000 and $13,000 in cash and, randomly, a boxing glove that Mr. Cohen said had been worn by a Brazilian mixed-martial arts fighter, Mr. Gauger...