She might have Mater going after her - that's old school! (4)
I believe the answer is:
alma
'she' is the definition.
'might have mater going after her that's old school' is the wordplay.
I cannot really see how this works, but
'mater' could be 'ma' (synonyms) and 'ma' is located in the answer.
This explanation may well be incorrect...
Can you help me to learn more?
(Other definitions for alma that I've seen before include "The . . . . Mater is the old school" , "One's university" , "Loving girl" , "dancing girl" , "Woman" .)