Persuasive conversation turning stale, developer of vaccine has admitted (5,4)
I believe the answer is:
sales talk
'persuasive conversation' is the definition.
(talk intended to persuade you to buy something)
'turning stale developer of vaccine has admitted' is the wordplay.
'turning' is an anagram indicator.
'developer of vaccine' becomes 'Salk' (Jonas Salk, developer of polio vaccine).
'has admitted' is an insertion indicator (admit can mean to bring in).
'stale' anagrammed gives 'alest'.
'alest' put within 'salk' is 'SALES-TALK'.
(Other definitions for sales talk that I've seen before include "Pitch" , "Persuasive material" , "promotional activity" , "lure to customer" .)