I always read that White wanted Moscone dead, and Milk happened to be in the office at the time.
Moscone initially agreed to White's request, but later refused the appointment at the urging of Milk and others. On November 27, 1978, White visited San Francisco City Hall with the later-declared intention of killing not only Moscone and Milk, but also two other San Francisco politicians, California Assembly Speaker and later S.F. mayor Willie Brown, and Supervisor Carol Ruth Silver, both of whom he also blamed for lobbying Moscone not to re-appoint him.[7] He arrived that day by climbing through a first-floor window on the side of City Hall carrying a .38 revolver and 10 rounds of ammunition. By entering the building through the window, White was able to avoid the recently installed metal detectors. After entering Moscone's office, White pleaded to be re-instated as supervisor, but Moscone said no. White then killed Moscone by shooting him in the shoulder and chest, and twice in the head. He then walked to the other side of City Hall to Milk's office, reloaded the gun, and fatally shot Milk five times, the final two shots fired with the gun's barrel touching Milk's skull, according to the medical examiner. White then fled City Hall, turning himself in at the San Francisco's Northern Police Station where he had been a police officer.
It sounds like White wanted to kill Moscone AND Milk, and 2 other people, one being Willie Brown.