Annoy for a bad purpose (6)
I believe the answer is:
offend
'annoy' is the definition.
(I've seen this before)
'a bad purpose' is the wordplay.
'a bad' becomes 'off' (eg meat that's gone off is bad).
'purpose' becomes 'end' (I've seen this before).
'off'+'end'='OFFEND'
'for' acts as a link.
(Other definitions for offend that I've seen before include "Be disagreeable to" , "Cause displeasure or hurt, be nasty towards" , "Commit an illegal act" , "Insult - sin" , "Be displeasing to" .)