Youngster is quaint on horse, that girl heading off (9)
I believe the answer is:
tweenager
'youngster' is the definition.
(child beginning or about to begin their teen years)
'quaint on horse that girl heading off' is the wordplay.
'quaint' becomes 'twee' (I've seen this before).
'on horse' becomes 'nag' (nag is a kind of horse. I am not sure about the 'on' bit.**).
'that girl' becomes 'her'.
'heading off' means to remove the first letter.
'her' with its initial letter taken away is 'er'.
'twee'+'nag'+'er'='TWEENAGER'
'is' acts as a link.
(Other definitions for tweenager that I've seen before include "Possibly a ten-year-old" , "Precocious child" , "Youngster" , "exasperating twelve-year-old" , "(Informally) a child approaching the age of 13?" .)