"Taming of the Shrew" role (4)
I believe the answer is:
'taming of the shrew role' is the definition.
(I've seen this in another clue)
This is the entire clue.
(Other definitions for kate that I've seen before include "Katherine (dim.)" , "'Kiss me, ---' (Shrew)" , "'Kiss me ___' (show)" , "Cole Porter musical 'Kiss Me ....'" .)