Revealing mistake: When Chekov and Terrell are walking on Ceti Alpha V watch as they get to the ridge overlooking the Botany Bay. Each of them cast two very distinct shadows, indicating two separate light sources. The planet only has one sun. (00:15:05)
Revealing mistake: When the Enterprise is leaving space dock, look at the right (starboard) side of the ship. You can see the shape of the mounting arm the model was mounted on, even though it was blacked out to look like space, and it also blocked out the bottom of the dock. The footage was reused from Star Trek: The Motion Picture.
Revealing mistake: Near the end when Khan says "For hate's sake, I spit my last breath at thee", if you look near the bridge of his nose you can see the prosthetic of his "wounds" is coming loose.
Revealing mistake: When Chekhov and Terrell see the Ceti Eels moving around in the tank for the first time, it's obvious that it's not sand in the tank. As the creatures move underneath, all the 'grains' move in unison, and not a single grain of sand is disturbed nor do they tumble over one another.
Revealing mistake: On the Regula station, when Kirk, Saavik and McCoy set phasers to stun, McCoy is squeezing the trigger on his as the emitter light is lit. He is still holding it down as he turns to walk off.
Revealing mistake: After crippling the Enterprise, the Reliant comes around again. As it does, the star field behind it can be seen through the black area of the inside of the nacelle.
Answer: At the time of the attack, most of Scotty's crew were attempting to flee a coolant leak caused by the damage. Cadet Preston (Scotty's nephew) remained at his post in the confusion, and was the reason the Enterprise was able to maintain minimal power with the energizers knocked out. Although he was wounded from the assault while rescuing another crew member, the coolant leak is what led to his ultimate demise, as the coolant was highly toxic. When Scotty brought him to the Bridge, he was himself quite disorientated from the events and was also devastated by his nephew's condition. Agreed it makes little to no sense, but Scotty didn't know what he was doing. His nephew was going to die, regardless of where he took him to. Going to sickbay would not have prevented it at this point. It seems like it was used as a mood breaker for the scene, crews celebrating their escape from death only to be confronted by it again in another form.