The cheek of judge, a wise old bird (4)
I believe the answer is:
jowl
'the cheek' is the definition.
(jowl can mean part of a cheek)
'judge a wise old bird' is the wordplay.
'judge' becomes 'j' (abbreviation for judge).
'a wise old bird' becomes 'owl' (owls are said to be wise).
'j'+'owl'='JOWL'
'of' acts as a link.
(Other definitions for jowl that I've seen before include "Drooping cheek" , "Pendulous double chin" , "Loose skin on a person's throat" , "Jaw - cheek" , "Fleshy cheek" .)