Influence of converted priest, say, is mounting (8)
I believe the answer is:
prestige
'influence' is the definition.
(thesaurus)
'converted priest say is mounting' is the wordplay.
'converted' is an anagram indicator.
'say' becomes 'eg'.
'is mounting' says the letters should be written backwards (going up in a down clue).
'eg' in reverse letter order is 'ge'.
'priest' with letters rearranged gives 'presti'.
'presti'+'ge'='PRESTIGE'
'of' is the link.
(Other definitions for prestige that I've seen before include "It's good standing" , "[GLAMOUR]" , "Kudos" , "High esteem" , "High standing, influence" .)