Stay and prosper, having refused Oscar (5)
I believe the answer is:
dwell
'stay' is the definition.
(I've seen this before)
'prosper having refused oscar' is the wordplay.
'prosper' becomes 'dowell'.
'having refused' indicates named letters should be taken away.
'oscar' becomes 'o' (phonetic alphabet: alpha, bravo, charlie etc.).
'dowell' with 'o' removed is 'DWELL'.
'and' is the link.
(Other definitions for dwell that I've seen before include "stay" , "Live (in a place)" , "Settle" , "Reside - linger" , "Remain" .)