Wife greeting Papa with favourite dog (7)
I believe the answer is:
whippet
'dog' is the definition.
(whippet is a kind of dog)
'wife greeting papa with favourite' is the wordplay.
'wife' becomes 'w'.
'greeting' becomes 'hi' (hi is a kind of greeting).
'papa' becomes 'p' (phonetic alphabet: alpha, bravo, charlie etc.).
'with' is a charade indicator (letters next to each other).
'favourite' becomes 'pet' ('pet' can be a synonym of 'favourite').
'w'+'hi'+'p'+'pet'='WHIPPET'
(Other definitions for whippet that I've seen before include "Greyhound relative" , "racing animal" , "Small slender breed of dog like a greyhound" , "Dog bred for racing" , "Racing-dog" .)