Cut and run, nabbed by police officer (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)
'run nabbed by police officer' is the wordplay.
'run' becomes 'r' (cricket abbreviation).
'nabbed by' means one lot of letters goes inside another (nab can mean to steal or take possession of).
'police officer' becomes 'cop' (cop is a kind of police officer).
'r' inserted within 'cop' is 'CROP'.
'and' is the link.
(Other definitions for crop that I've seen before include "Cut short the harvest" , "Graze" , "The handle of a whip, or part of bird's neck" , "Harvest - short hairstyle - whip" , "Set" .)