Finished the port duke left out (4)
I believe the answer is:
over
'finished' is the definition.
('over' can be similar in meaning to 'finished')
'the port duke left out' is the wordplay.
'the port' becomes 'dover' (I've seen this before).
'duke' becomes 'd'.
'left out' suggests deleting specific letters.
'dover' with 'd' removed is 'OVER'.
(Other definitions for over that I've seen before include "Ancient history" , "end to radio communication?" , "jumped fence" , "settled" , "Half a dozen deliveries" .)