One trying to hear register with French name coming up (8)
I believe the answer is:
listener
'one trying to hear' is the definition.
(I've seen this before)
'register with french name coming up' is the wordplay.
'register' becomes 'list' (listing is a kind of registering).
'with' is a charade indicator (letters next to each other).
'french name' becomes 'rene' (I've seen this in another clue).
'coming up' says the letters should be written backwards (in a down clue, letters go up).
'rene' back-to-front is 'ener'.
'list'+'ener'='LISTENER'
(Other definitions for listener that I've seen before include "Eavesdropper" , "Auditor" , "One enjoying radio broadcast" , "Dead body" , "One trying to hear" .)