WASHINGTON—The botched bombing last month of a Yemeni school bus by a Saudi-led military coalition has ignited a new debate in the Trump administration about how much military support the U.S. should provide its Gulf allies in the three-year-old conflict.

Competing factions within the administration are at odds over the best way to prevent more of the errant airstrikes by Saudi Arabia and the United Arab Emirates that have killed thousands of civilians in Yemen.

According...