Cut and run? Not quite (3)
I believe the answer is:
lop
'cut' is the definition.
(I know that lop is a more specific form of the action thin out)
'run? not quite' is the wordplay.
'run?' becomes 'lope' (lope is a kind of run).
'not quite' means to remove the last letter (most but not all of the word).
'lope' with its final letter taken off is 'LOP'.
'and' is the link.
(Other definitions for lop that I've seen before include "Hack" , "Prune, cut the top off" , "Cut away branches or twigs" , "Remove branches from a tree" , "Cut (e.g. branch off)" .)