CANBERRA, Australia—A leaked recording emerged of Australia’s prime minister mocking President Donald Trump, the latest awkward episode between two allied leaders whose relationship had gotten off to a rocky start.

In an address to an annual press ball at Australia’s Parliament on Wednesday, Prime Minister Malcolm Turnbull poked fun at the U.S. president about his alleged connections to Russia, and over their first official meeting last month on a mothballed aircraft carrier in New York.

“It...