Religious book very fortunate to lose its covers (4)
I believe the answer is:
luke
'religious book' is the definition.
(Gospel of Luke)
'very fortunate to lose its covers' is the wordplay.
'very fortunate' becomes 'flukey' (a fluke is a bit of good luck).
'to lose its covers' indicates the central letters (the letters which cover the rest are removed).
The middle letters of 'flukey' are 'LUKE'.
(Other definitions for luke that I've seen before include "One of the Apostles" , "Gospel writer" , "Patron saint of physicians" , "Third NT book" , "One of the four Evangelists" .)