Pub getting in cut fish (6)
I believe the answer is:
minnow
'fish' is the definition.
(minnow is a kind of fish)
'pub getting in cut' is the wordplay.
'pub' becomes 'inn' (an inn is a type of pub).
'getting in' is an insertion indicator.
'cut' becomes 'mow' (as in mowing a lawn).
'inn' inserted within 'mow' is 'MINNOW'.
(Other definitions for minnow that I've seen before include "unimportant person" , "Little person" , "Small river fish" , "Tiny freshwater fish" , "nobody" .)