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 anagramming the letters.
'wasnt' with its last letter taken off is 'wasn'.
'wasn' is an anagram of '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?" .)