Racing dog beat favourite (7)
I believe the answer is:
whippet
'racing dog' is the definition.
(I've seen this before)
'beat favourite' is the wordplay.
'beat' becomes 'whip' (whipping is a kind of beating).
'favourite' becomes 'pet' (synonyms).
'whip'+'pet'='WHIPPET'
(Other definitions for whippet that I've seen before include "Dog related to 9 down" , "racing animal" , "Fast runner" , "dog class?" , "Small greyhound-type dog" .)