Object to being given another errand? (6)
I believe the answer is:
resent
I believe this clue is a double definition.
'object to' is the first definition.
(to resent something is to dislike it)
'given another errand?' is the second definition.
(as in to send someone on an errand)
'being' is the link.
(Other definitions for resent that I've seen before include "being put in post again" , "Consider as an injury or affront" , "Feel bad about" , "Take offence at" , "Dispatched again - take amiss" .)