Priest and waggish faculty head keeping company (10)
I believe the answer is:
archdeacon
'priest' is the definition.
(type of clergyman)
'waggish faculty head keeping company' is the wordplay.
'waggish' becomes 'arch' (both can mean sarcastic or playful).
'faculty head' becomes 'dean' (head of a university department).
'keeping' indicates putting letters inside.
'company' becomes 'co' (abbreviation).
'dean' enclosing 'co' is 'deacon'.
'arch'+'deacon'='ARCHDEACON'
'and' is the link.
(Other definitions for archdeacon that I've seen before include "Senior Christian cleric" , "Member of clergy" , "Church dignitary" , "Anglican clergyman" , "Church official" .)