Annoy by leaving on purpose (6)
I believe the answer is:
offend
'annoy' is the definition.
(I've seen this before)
'leaving on purpose' is the wordplay.
'leaving' becomes 'off' (I've seen this in other clues).
'on' means one lot of letters go next to another.
'purpose' becomes 'end' (I've seen this before).
'off'+'end'='OFFEND'
'by' is the link.
(Other definitions for offend that I've seen before include "Commit an illegal act" , "disgust" , "Distress" , "Wound (one's) feelings" , "Cause displeasure to, insult" .)