Continuity mistake: When Kirk splits up the landing party, he asks McCoy to further inspect the vegetation. In the two shot, McCoy isn't wearing his tricorder, but in close-up, it's there, strapped over his left shoulder. Cut back to the two shot, though, and it's gone again.
Continuity mistake: The Providers' representative colors on the game platform and the thralls' collars (all three colors are on Galt's collar) are red, yellow and blue. But when Kirk appears before them, the Providers themselves are red, yellow and green, and so is the gaming symbol inside their glass bubble.
Continuity mistake: Kirk's arrow strikes the Klingon in the knee. But a few shots later, the shaft is protruding from his thigh, several inches above its original position. Still later, he's holding one hand to his calf just below the knee, which isn't where the arrow was either time.
Continuity mistake: During the scene when the Klingon agent is exposed by the Tribbles, Kirk first holds the two furballs as if they had a handle on the underside (probably for the parts to make them wriggle). As he turns to Spock (the wide-angle shot), he holds the Tribbles cupped in his hands, his thumbs supporting their flanks. But as Kirk approaches Spock (close-up), he has them 'by the handle' again.
Factual error: When the Enterprise is slingshotting its way back to the "present" it uses the computer to deposit the two men at different times, The Enterprise is traveling backwards in time, so will logically meet the time when the guard was beamed up first, not after Christopher is returned to his cockpit. So the order is wrong. Should be guard then Christopher, not Christopher then guard.
Continuity mistake: As Mira is speaking with the alien's voice, Spock is behind a console that has a piece of equipment attached to it. From behind Spock, the piece is to his left, but the close ups of Spock show him close to and directly in front of it.