Chaplain having a quiet read, possibly (5)
I believe the answer is:
padre
'chaplain' is the definition.
(padre is a kind of chaplain)
'a quiet read possibly' is the wordplay.
'a quiet' becomes 'p' (music symbol for 'piano'**).
'possibly' indicates an anagram.
'read' is an anagram of 'adre'.
'p'+'adre'='PADRE'
'having' acts as a link.
(Other definitions for padre that I've seen before include "clerical worker" , "Chaplain to armed forces" , "A military clergyman" , "Army chaplain" , "Army serviceman?" .)