WASHINGTON—President Trump signed an agreement with Poland to send 1,000 additional U.S. troops to Poland while treating his visiting Polish counterpart to a military flyover at the White House as thanks for a commitment to buy F-35 jet fighters. Since 2017, the U.S. has kept a steady rotation of about 4,000 troops in the Polish countryside. The new troops wouldn’t be combat troops but rather “enabling forces” with jobs such as manning drones and undertaking logistics. A timetable for the troop arrival wasn’t announced, and Mr. Trump said forces could be taken from other bases in Europe, specifically mentioning Germany, which, like Poland, is a North Atlantic Treaty Organization ally. Shortly before a Rose Garden news conference, Mr. Trump and the Polish President stood on the White House South Lawn and watched as an F-35 flew over, a show of gratitude for Polish plans to purchase more than 30 of the Lockheed Martin Corp.-made fighters.