Old monk’s sleeping-place at end of compline (4)
I believe the answer is:
bede
'old monk's' is the definition.
(I've seen this before)
'sleeping-place at end of compline' is the wordplay.
'sleeping place' becomes 'bed' (I've seen this before).
'at end of' indicates one should take the final letters.
The final letter of 'compline' is 'e'.
'bed'+'e'='BEDE'
(Other definitions for bede that I've seen before include "Authority consulted on English history" , "He wrote the eighth century Ecclesiastical History of the English Nation" , "Northumbrian monk and historian, d. 735" , "old monk" , "Venerable author of The Ecclesiastical History of the English People, d. 735" .)