Eastern racecourse, not a current course (6)
I believe the answer is:
entree
'a current course' is the definition.
'entree' can be an answer for 'course' (course of a meal). I'm not certain of the 'a current' bit.
'eastern racecourse not' is the wordplay.
'eastern' becomes 'e'.
'racecourse not' becomes 'ntree' (I can't justify this - if you can you should believe this answer much more).
'e'+'ntree'='ENTREE'
(Other definitions for entree that I've seen before include "Dish providing right of entry" , "Access - course at meal" , "Right to take part" , "The right to be admitted" , "Starter" .)