Boxer with cut covering up, tho' woozy (8)
I believe the answer is:
southpaw
'boxer' is the definition.
(term for a left-handed boxer)
'cut covering up tho' woozy' is the wordplay.
'cut' becomes 'saw' (sawing is a kind of cutting).
'covering' means one lot of letters goes inside another.
'woozy' indicates an anagram (I've seen this in other clues).
'up'+'tho'='uptho'
'uptho' anagrammed gives 'outhp'.
'saw' going around 'outhp' is 'SOUTHPAW'.
'with' acts as a link.
(Other definitions for southpaw that I've seen before include "Left-handed pugilist" , "To wash up (anag.)" , "Boxer who leads with right fist" , "Left-handed person, especially a boxer" , "A left-handed boxer who leads with the right" .)