U.S. military officials sought permission to send an armed drone near a patrol of Green Berets before a deadly ambush Oct. 4 in Niger, but the request was blocked, raising questions about whether those forces had adequate protection against the dangers of their mission.

New information shows the Green Beret team was part of a larger mission, one potentially more dangerous than initially described, and one believed to merit an armed drone. But the request was blocked in a chain of approval that snakes through the Pentagon,...