Approve though possibly sore after the finish (7)
I believe the answer is:
endorse
'approve' is the definition.
(endorse can mean to approve or support)
'possibly sore after the finish' is the wordplay.
'possibly' indicates an anagram.
'after' means one lot of letters go next to another.
'the finish' becomes 'end' (I've seen this in other clues).
'sore' anagrammed gives 'orse'.
'orse' put after 'end' is 'ENDORSE'.
'though' is the link.
(Other definitions for endorse that I've seen before include "Corroborate" , "Red ones (anag.)" , "Back, sanction" , "Approve - support" , "Give backing to" .)