Girl with a car at finishing school (5)
I believe the answer is:
carol
'girl' is the definition.
(woman's name)
'with a car at finishing school' is the wordplay.
'with' says to put letters next to each other.
'a car at' becomes 'caro' (I can't justify this - if you can you should give a lot more credence to this answer).
'finishing' says to take the final letters (the letter which finishes the word).
The last letter of 'school' is 'l'.
'caro' put next to 'l' is 'CAROL'.
Can you help me to learn more?
(Other definitions for carol that I've seen before include "sing happily" , "See 1" , "Girl" , "Song, especially at Christmas" , "Girl's name; sing" .)