RIYADH, Saudi Arabia—Secretary of State Mike Pompeo met with Saudi Arabia’s leaders on Monday, bearing a message of support for a close Trump administration ally at a time when the relationship is under pressure at home and abroad.

Mr. Pompeo met separately with King Salman and Crown Prince Mohammed bin Salman at the royal court in the kingdom’s capital, telling reporters afterward that he raised at least two difficult issues—the war in Yemen and the 2018 killing of Saudi journalist Jamal Khashoggi at Saudi Arabia’s consulate...