Domestic animal hospital admits a bishop for head cold (7)
I believe the answer is:
catarrh
'head cold' is the definition.
(I've seen this before)
'domestic animal hospital admits a bishop' is the wordplay.
'domestic animal' becomes 'cat' (I've seen this before).
'hospital' becomes 'h'.
'admits' is an insertion indicator.
'bishop' becomes 'rr' (abbreviation for Right Reverend).
'cat'+'h'='cath'
'a'+'rr'='arr'
'cath' placed around 'arr' is 'CATARRH'.
'for' acts as a link.
(Other definitions for catarrh that I've seen before include "Nasal inflammation" , "Inflammation of membrane of nose and throat" , "cold symptom" , "Arch rat with inflamed mucous membrane" , "(Inflammation causing) excessive mucus" .)