Rash drive round old university (9)
I believe the answer is:
impetuous
'rash' is the definition.
(thesaurus)
'drive round old university' is the wordplay.
'drive' becomes 'impetus' (impetus is a kind of drive**).
'round' means one lot of letters goes inside another.
'old' becomes 'o' (common abbreviation eg in OE for Old English).
'university' becomes 'u'.
'o'+'u'='ou'
'impetus' going around 'ou' is 'IMPETUOUS'.
(Other definitions for impetuous that I've seen before include "Impulsive, hasty" , "Acting rashly" , "Hot-headed" , "Rash" .)