Consider an affront to be given another errand (6)
I believe the answer is:
resent
This clue is a double definition.
'consider an affront' is the first definition.
(to consider something an affront is to resent it)
'given another errand' is the second definition.
(as in to send someone on an errand)
'to be' is the link.
(Other definitions for resent that I've seen before include "grudge" , "Dislike" , "Feel bitter or indignant about" , "Take offence at" , "Consider as an injury or affront" .)