One graduate’s avoiding cake, a Cuban export (5)
I believe the answer is:
rumba
'a cuban export' is the definition.
(Cuban dance)
'one graduate's avoiding cake' is the wordplay.
'one graduate' becomes 'ba'.
'avoiding' suggests deleting specific letters.
'cake' becomes 'rumbaba' (rum baba is a kind of cake).
'rumbaba' with 'ba' taken away is 'RUMBA'.
(Other definitions for rumba that I've seen before include "Cuban folk-dance" , "number of steps" , "Latin American dance" , "Caribbean dance" , "South American dance" .)