Hasten current quietly flowing back in river (8)
I believe the answer is:
expedite
'hasten current' is the definition.
'expedite' can be an answer for 'hasten' (synonyms). I'm not certain of the 'current' bit.
'quietly flowing back in river' is the wordplay.
'quietly' becomes 'p' (musical symbol).
'flowing back' becomes 'edit' (I can't explain this - if you can you should believe this answer much more).
'in' means one lot of letters goes inside another.
'river' becomes 'exe' (River Exe).
'p'+'edit'='pedit'
'pedit' inserted within 'exe' is 'EXPEDITE'.
Can you help me to learn more?
(Other definitions for expedite that I've seen before include "Make happen faster" , "Push through faster" , "Hasten (process)" , "Hasten progress" , "Hasten, facilitate" .)