President Donald Trump’s personal attorney Michael Cohen helped a law and lobbying firm land a corporate client with ties to Kushner Cos., the family company of White House adviser Jared Kushner that is currently the subject of a federal probe.

Under a 2017 contract with Squire Patton Boggs, Mr. Cohen was paid $500,000 a year to help it drum up business, prosecutors said in a recent filing in federal court in New York, where Mr. Cohen is under investigation for potential bank fraud and campaign-finance violations. He also...