Almost wasn't upset when cut (4)
I believe the answer is:
sawn
'cut' is the definition.
(cut with a saw)
'almost wasn't upset' is the wordplay.
'almost' means to remove the last letter (most of the word but not all of it).
'upset' indicates an anagram.
'wasnt' with its last letter taken away is 'wasn'.
'wasn' with letters rearranged gives 'SAWN'.
'when' is the link.
(Other definitions for sawn that I've seen before include "Cut with serrated blade" , "timber must be" , "Severed" , "(Of wood) cut" , "went up and down?" .)