Finished without Rex however (6)
I believe the answer is:
though
'however' is the definition.
('though' can be a synonym of 'however')
'finished without rex' is the wordplay.
'finished' becomes 'through' ('through' can be similar in meaning to 'finished').
'without' suggests deleting specific letters.
'rex' becomes 'r' (common abbreviation e.g. GR=George Rex, King Georg).
'through' with 'r' removed is 'THOUGH'.
(Other definitions for though that I've seen before include "Nevertheless" , "Even if" , "Despite the fact that" , "Concessive conjuction" , "However" .)