Annoy by no longer working to the finish (6)
I believe the answer is:
offend
'annoy' is the definition.
(I've seen this before)
'by no longer working to the finish' is the wordplay.
'by' says to put letters next to each other.
'no longer working' becomes 'off' (I can't explain this - if you can you should give a lot more credence to this answer).
'to the finish' becomes 'end' (I've seen this in another clue).
'off' put next to 'end' is 'OFFEND'.
Can you help me to learn more?
(Other definitions for offend that I've seen before include "Annoy - upset" , "Wound (one's) feelings" , "Insult - sin" , "Commit an illegal act" , "Break the rules" .)