Chief almost makes a mistake (5)
I believe the answer is:
gaffe
'a mistake' is the definition.
(gaffe is a kind of mistake)
'chief almost' is the wordplay.
'chief' becomes 'gaffer' ('gaffer' can be a synonym of 'chief').
'almost' means to remove the last letter (most of the word but not all of it).
'gaffer' with its final letter taken off is 'GAFFE'.
'makes' is the link.
(Other definitions for gaffe that I've seen before include "Brick (that's dropped)" , "Blundering remark" , "Remark one wishes unmade" , "A foolish error" , "Clanger" .)