Cut and run, almost (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 almost' is the wordplay.
'run' becomes 'lope' (loping is a kind of running).
'almost' means to remove the last letter (most of the word but not all of it).
'lope' with its last letter taken off is 'LOP'.
'and' is the link.
(Other definitions for lop that I've seen before include "Cut off, tree branches say" , "Hack" , "Prune, cut the top off" , "Cut (e.g. branch off)" , "Remove branches from a tree" .)