Goofs

When Elaan throws her dagger at Kirk, it barely misses his shoulder and instead gets lodged into a piece of art on the wall. In later scenes in Elaan's quarters, the piece of art is still present, but there's no hole where the dagger was stuck.

When the Klingon ship attacks, Chekov serves as the right-hand navigator. But when we switch to Uhura's POV, a taller blond man is suddenly in his place. This is stock footage where Sulu is also seen turning in his seat to look at the Captain, and then returns to look forward towards the view screen.

(at around 15 mins) When Kirk responds to the intercom hail from Security reporting a disturbance in the Dohlman's quarters, he is clearly located at Spock's science station. Note the audio speaker, tape player, array of five double switches and earpiece. The subsequent mid-conversation wide shot shows Kirk (and Spock) standing at Kirk's command chair.

Near the end of the scene as the Kingons are coming in for their final attack, as Capt Kirk says "Chekov, photon torpedoes, fire!" In the background behind and to the right of Kirk you can see Elaan sitting in a chair with Scotty standing next to her, and you see the camera caught France Nuyen breaking character looking at or towards Scotty to her right smiling, as if she was reacting to something funny being said just before the camera started filming and then, still smiling, looking towards her left.

When Elaan throws her knife at Kirk, its guide wire is visible as it hits the wall next to him.

According to the description of the sabotage performed on Enterprise's power system, plus what is shown, the Enterprise maintains main power using a single dilithium crystal and has no replacements on board in case it is damaged. Such a nonsensical design could never be approved for use aboard a ship of the line, especially not one with the responsibilities such as Enterprise holds. The risk of being disabled or destroyed would be far too great to make it feasible.

Elaan is housed in Uhura's quarters. Since it is well established that the Enterprise has plenty of guest rooms for VIP passengers (see, e.g., Journey to Babel (1967)), it would seem unnecessary that a junior officer would have to give up her quarters for a single visiting dignitary.

The Klingon ship is following Enterprise, which is traveling on impulse power, no farther than 500,000 kilometers away, yet when Sulu says that the Klingon ship is approaching Enterprise "at warp speed", Capt. Kirk has enough time to get back to the bridge before the Klingon ship is even within phaser range. This makes no sense, as the Klingon ship would have long overshot Enterprise in seconds if it had been approaching even at warp 1.

Mr. Sulu then says the Klingon ship is traveling at warp 6, but at this speed, the Klingon ship would overshoot Enterprise by millions of kilometers within a few seconds.

Also, the Klingon ship is pictured on the viewscreen and can be seen maneuvering at supposedly warp speeds. Warp 1 is the speed of light, so if the Klingon ship were, indeed, traveling at warp speed, its movement would not be visible to the human eye.

As soon as Kirk realized that Enterprise was being shadowed by a Klingon cruiser, he should have broken radio silence and advised Starfleet Command. His orders to maintain radio silence were to keep the Klingons from finding out that a Federation vessel was operating in the system. Since the Klingons have a ship shadowing Enterprise, they obviously know she's there. He wouldn't have had to call for help; he should have advised Starfleet Command that the 'cat was out of the bag'.

The Organian Peace Treaty (or perhaps even the Organians themselves) should've prevented the Klingons unprovoked attack on the Enterprise.

At one point in her training, Elaan says "If I have to stay here for 10 light-years, I will not be soiled with any contact with you." She makes the common mistake of treating light-years as a unit of time, when it is a unit of distance.

The Dohlman of Elas is guided to a seat by an engineer while on the bridge. He grabs her arm. Earlier The Dohlman of Elas attacked Kirk and others when she is touched.