Everyman's beginning to go under: a quiet bit of snoozing in part of church (4)
I believe the answer is:
apse
'part of church' is the definition.
(part of a church building)
'everyman's beginning to go under a quiet bit of snoozing' is the wordplay.
'beginning' says to take the initial letters.
'to go under' means one lot of letters go next to another (in a down clue, some letters go below others).
'quiet' becomes 'p' (music symbol for 'piano', ie soft or quiet).
'bit of' says to take the initial letters.
The initial letter of 'everymans' is 'e'.
The initial letter of 'snoozing' is 's'.
'a'+'p'+'s'='aps'
'e' after 'aps' is 'APSE'.
'in' is the link.
(Other definitions for apse that I've seen before include "Projection to the east end of a church" , "Church recess with domed or arched roof" , "Semicircular recess in church" , "Where the altar is usually found in a church" , "place in church" .)