German supply dumps in the forward areas were repeatedly hit by South African Marylands, flying under escort. Dumps near Gambut, between Tobruk and the border were singled out for more attention. The result was that German preparations for a renewed attack on Tobruk were delayed, so that the British were ready for offensive operations before Rommel.
Besides supply lines, the battle for air supremacy was waged by a bombing campaign against Axis airfields. Fields at Gambut and Gazala were particularly targeted. Gazala was the main fighter base.
British fighter aircraft were totally devoted to air defense of the main army bases and convoys to Tobruk, and to escorting reconnaissance and bomber aircraft. The alternate, more offensive strategy would have entailed patrols over Axis-held territory, looking for opposing aircraft to engage.
This is based on the account in the Official History.