Little known fact............
As scripted AND filmed--when Spock goes back to clear the affected "tube" he is stung by a hornet on his right index finger. The hornet's nest was in fact what caused the tube to malfunction.
Both before and after Nimoy goes into the engine area off camera--his right index finger sports some makeup and prosthetics of a hugely swollen finger.
In the early section of the scene he simply holds it off camera--in the later portion of the scene it can be glimpsed a few times.
The bigger the TV you are watching it on the easier it is to see it. I am watching a 32" set right now and can see it. People with larger sets can see it more clearly for sure.
At the split second before Kirk turns from the main console to the rear controls, Spock is in fact holding up the finger for Kirk to see.
You got to be handy with the pause button, but there's no doubt.