She has the Eastern rival beaten (6)
I believe the answer is:
elvira
'she' is the definition.
'the eastern rival beaten' is the wordplay.
'the eastern' becomes 'e'.
'beaten' is an anagram indicator (letters beaten up or destroyed).
'rival' anagrammed gives 'lvira'.
'e'+'lvira'='ELVIRA'
'has' acts as a link.
(Other definitions for elvira that I've seen before include "Coward spirit" , "Girl's name (from Don Giovanni)" .)