Quietly goes ahead with appeals (6)
I believe the answer is:
pleads
'appeals' is the definition.
(pleading is a kind of appealing)
'quietly goes ahead' is the wordplay.
'quietly' becomes 'p' (p for piano and quiet).
'goes ahead' becomes 'leads' (I've seen this before).
'p'+'leads'='PLEADS'
'with' acts as a link.
(Other definitions for pleads that I've seen before include "Implores" , "Begs" , "Appeals or requests earnestly" , "Makes appeal or entreaty" .)