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 definition 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 explain this - if you can you should give a lot more credence to this answer).
'miss'+'al'='MISSAL'
'giving' is the 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" .)