Old president pursues black cutie (4)
I believe the answer is:
babe
'cutie' is the definition.
(thesaurus)
'old president pursues black' is the wordplay.
'old president' becomes 'abe' (Abraham (Abe) Lincoln).
'pursues' is a charade indicator (letters next to each other).
'black' becomes 'b' (as in pencil classifications).
'abe' after 'b' is 'BABE'.
(Other definitions for babe that I've seen before include "Tiny infant or good-looking girl" , "Girlfriend" , "Stunner" , "Darling" , "Youngster" .)