Job centre overwhelms her at first (5)
I believe the answer is:
chore
'job' is the definition.
(synonyms)
'centre overwhelms her at first' is the wordplay.
'centre' becomes 'core' (core can mean something's centre).
'overwhelms' is an insertion indicator.
'at first' indicates taking the first letters.
The first letter of 'her' is 'h'.
'core' enclosing 'h' is 'CHORE'.
(Other definitions for chore that I've seen before include "A household or routine task" , "Boring job" , "Stream's murmuring" , "Unpleasant task" , "Drudgery" .)