Rival's advance to embrace darling (7)
I believe the answer is:
compete
'rival's' is the definition.
(thesaurus)
'advance to embrace darling' is the wordplay.
'advance' becomes 'come' (I've seen this in other clues).
'to embrace' is an insertion indicator.
'darling' becomes 'pet' ('pet' can be a synonym of 'darling').
'come' enclosing 'pet' is 'COMPETE'.
(Other definitions for compete that I've seen before include "Vie, contest" , "Take part in contest" , "Contend (for a prize)" , "Struggle for victory" , "Strive (against others)" .)