Chief almost made a mistake (5)
I believe the answer is:
gaffe
'a mistake' is the definition.
(a gaffe is a mistake or error)
'chief almost' is the wordplay.
'chief' becomes 'gaffer' (synonyms).
'almost' means to remove the last letter (almost the whole word is kept).
'gaffer' with its final letter taken away is 'GAFFE'.
'made' acts as a link.
(Other definitions for gaffe that I've seen before include "Social blunder" , "Something wrong" , "Tactless remark" , "bad mistake" , "Brick (that's dropped)" .)