Cool headed girl with quality (5)
I believe the answer is:
class
'quality' is the definition.
(class is a kind of quality)
'cool headed girl' is the wordplay.
'cool' becomes 'cl' (I am not sure about this - if you are sure you should give a lot more credence to this answer).
'headed' means to remove the first letter (head can mean to remove the top from something).
'girl' becomes 'lass' (lass can informally mean a girl).
'lass' with its first letter taken off is 'ass'.
'cl'+'ass'='CLASS'
'with' is the link.
Can you help me to learn more?
(Other definitions for class that I've seen before include "Set, category" , "Order - quality" , "Category; form" , "Category; lesson" , "pupils?" .)