Over drug, rising reputation's at issue (7)
I believe the answer is:
emanate
'issue' is the definition.
(thesaurus)
'over drug rising reputation's' is the wordplay.
'over' is a reversal indicator.
'drug rising' becomes 'eta' (I can't justify this - if you can you should give a lot more credence to this answer).
'reputation' becomes 'name' (name is a kind of reputation).
'eta'+'name'='etaname'
'etaname' written backwards gives 'EMANATE'.
'at' is the link.
Can you help me to learn more?
(Other definitions for emanate that I've seen before include "Send out" , "Come out (of source)" , "Issue or originate from manatee" , "Emerge or originate" , "spring" .)