Put up with something hairy for the most part (4)
I believe the answer is:
bear
'put up with' is the definition.
(synonyms)
'something hairy for the most part' is the wordplay.
'something hairy' becomes 'beard' (facial hair).
'for the most part' means to remove the last letter.
'beard' with its last letter taken away is 'BEAR'.
(Other definitions for bear that I've seen before include "It may be polar or grizzly" , "Support (a weight)" , "Carry; large mammal" , "Huge carnivore" , "ill-mannered person" .)