Nick accepts barber's final cut (4)
I believe the answer is:
crop
'cut' is the definition.
(I know that crop is a more specific form of the action the thin out)
'nick accepts barber's final' is the wordplay.
'nick' becomes 'cop' (both can mean to apprehend or arrest).
'accepts' means one lot of letters goes inside another.
'final' suggests the final letters.
The final letter of 'barber' is 'r'.
'cop' going around 'r' is 'CROP'.
(Other definitions for crop that I've seen before include "Graze" , "(Yield from) cultivated plant" , "Expanded part of bird's throat" , "It's produced at harvest time" , "Yield -- trim" .)