Lively daughter returned, following in fast vehicle (4,3)
I believe the answer is:
indy car
'fast vehicle' is the definition.
'lively daughter returned following in' is the wordplay.
'lively' becomes 'racy' (I've seen this before**).
'daughter' becomes 'd'.
'returned' says the letters should be written in reverse.
'following' says to put letters next to each other.
'racy'+'d'='racyd'
'racyd' reversed gives 'dycar'.
'dycar' after 'in' is 'INDY CAR'.
(Other definitions for indy car that I've seen before include "High-speed racer" , "US motor" .)