Continuity mistake: Just after the ship is first attacked, Scotty tells McCoy, "We can't fire full phasers with our screens up." Say what? The Enterprise couldn't operate transporters with the shields up, but it fired its phasers, full or otherwise, with the screens up every time it went into battle, and always had. If they'd been forced to shut their shields down every time they fired, the Enterprise would have been history long before this.
Plot hole: The ambassador and his aide beam down to the planet. This would not be possible because earlier on Scotty refused to lower the screens/shields until the Captain told him to do so. It had already been established in an earlier episode titled "Arena" that transporters don't work with screens up.
Continuity mistake: The landing party had been apprehended and placed in a locked room under the supervision of a single guard. Spock manages to free them and Kirk knocks out this guard, leaving him in the locked room. Later on in the episode, they return to this very room and somehow this guard is gone. He could not have woken up and walked out because it is a locked room and Kirk took his key.