Finish off fleshy fruit (4)
I believe the answer is:
plum
'fruit' is the definition.
(plum is a kind of fruit)
'finish off fleshy' is the wordplay.
'finish off' means to remove the last letter.
'fleshy' becomes 'plump' (plump can mean fleshy or fat).
'plump' with its last letter taken away is 'PLUM'.
(Other definitions for plum that I've seen before include "Oval fruit with single hard stone" , "superior kind?" , "Brilliant" , "Fruit < that's highly desirable" , "Select" .)