Spock had been injured (presumably) recently on Rigel VII. ("Three people killed--including Pike's yeoman, and seven injured.") Other notable injuries include Jose Tyler (who has his hand bandaged) and the unnamed geologist (who has a bandage on his neck).
I must go back and rewatch, again.
I don't remember the hand bandage appearing until after they direct the ships power against the elevator in the hill.
You can see Jose Tyler's bandaged hand starting with the very first shot he appears in on the bridge. But here's a shot as the landing party is beaming down for the first time (certainly well before the laser cannon is used on the rocky knoll). You can see Tyler (way over at screen left) has his right hand wrapped with a flesh-colored ACE bandage, and the Geologist way over at screen right has a white bandage on the left side of his neck.