The girl I'm giving back the book to (6)
I believe the answer is:
missal
'back the book to' is the definition.
I can't judge whether this defines the answer.
'the girl i'm' is the wordplay.
'the girl' becomes 'miss' (term of address for an unmarried woman).
'i'm' becomes 'al' (I can't justify this - if you can you should believe this answer much more).
'miss'+'al'='MISSAL'
'giving' acts as a link.
Can you help me to learn more?
(Other definitions for missal that I've seen before include "Church book covering all services in the year" , "Service-book" , "Prayer-book" , "Book of prayers and rites" , "Book used at Catholic mass" .)