Sean Hannity has been the king of Fox News for two and a half years, the highest-rated host on the highest-rated cable news channel. His dominance has withstood network management changes and the vicissitudes of a news cycle driven by President Trump, the anchor’s close political ally. In April, however, Mr. Hannity slipped back. For the first time since moving to the 9 p.m. time slot in October 2017, “Hannity” ranked second in Fox News’s prime-time lineup. “Tucker Carlson Tonight” ranked ahead, with an average nightly audience of 4.56 million viewers, outpacing the 4.39 million who tuned in for Mr....