Continuity mistake: After Garth attempts to convince Mr Scott that he is Captain Kirk and to beam him up, while he is banging his fists on the floor in anger he transforms back into himself and the fly like ring on his finger breaks and slides out of shot. It's quite hard to see but it does. The next time we see Garth's hand his ring is intact yet he has not transformed himself again to repair it.
Audio problem: Just as Garth brings Spock back and releases Kirk from the cell, "inviting" them to dinner, there's an odd buzzing/rattling noise in the audio that shouldn't be there. (00:11:30)
Continuity mistake: When Garth forces Kirk to kneel at his feet, Kirk's hands are extended palms down, then palms up, then palms down again as the camera angles change. (00:34:20)
Other mistake: At the end, Kirk teases Spock twice specifically about letting himself be hit on the head in order to determine which Kirk was genuine. But the phony Kirk never hit Spock on the head. He merely pushed Spock over and attacked the real Kirk. Clearly, the action sequence wound up differing from the scripted dialogue, and no one noticed. (Also clearly, Kirk is not speaking metaphorically here.) (00:45:30 - 00:49:20)
Continuity mistake: When Scott orders Sulu to fire phasers to punch through the force field, the two phaser beams are diverging when they leave the ship, but converging somehow (space mirrors?) when they strike the planet. (00:41:20)
Continuity mistake: As Garth brings in the chair and orders the governor's torture, the pedestal fruit bowl on the table in front of Kirk keeps moving back and forth in relation to him and to the wine pitcher. (00:22:40)
Revealing mistake: When Marta kisses Kirk, her hands leave green make-up smeared all over the left shoulder and back of his uniform tunic. (00:22:05)
Continuity mistake: After Marta's dance, Kirk and Spock are sitting an inch apart behind the table, then a foot apart, then an inch again and so forth every time the shot changes. (00:16:25)
Continuity mistake: When Scotty thwarts Garth, disguised as Kirk, attempts to beam to the Enterprise using the Queen-to-Queen level three code, Kirk throws a tantrum and accidentally loosens one of his controls during his rant right before he turns back to Garth.
Suggested correction: How is that a mistake?
Other mistake: The only way to beam up to the ship is to repeat the chess counter move after ASKING to be beamed aboard. After the two Captain Kirks fight ends with Spock stunning Garth with the phaser, Spock contacts the Enterprise. After saying "Spock to Enterprise, and hearing Scotty say back to him "Scott here", Scotty says" Queen to Queen's level 3" without Spock requesting to be beamed aboard or even waiting to see what Spock wanted in the first place. (00:46:56)