They're supposed to be seeking out new life and new civilizations
...But only in the opening credits, it seems. Starfleet PR Corps may be painting a rosier picture of the reality than the one the grunts in the front lines really face.
Because of the events of Space Seed, the ship was likely behind schedule.
That presents an interesting question: Kirk was traversing through space nobody from Earth or the Federation has any business traversing, not for the past few centuries anyway. Why? Was he taking a shortcut to Gamma 400, or was he actually tasked with checking up on this volume of space?
If the latter, then him filing a "All Quiet on the Mutara Front" report might be neglect of duty. Not because it hides the existence of Khan (which is fine and well) but because Kirk failed to conduct his scheduled mission because of Khan.
Timo Saloniemi