The Central Intelligence Agency has made a series of secret concessions in its drone campaign after military and diplomatic officials complained large strikes were damaging the fragile U.S. relationship with Pakistan.

The covert drones are credited with killing hundreds of suspected militants, and few U.S. officials have publicly criticized the campaign, or its rapid expansion under President Barack Obama. Behind the scenes, however, many key U.S. military and State Department officials demanded more-selective strikes. That...