Speak in public close to hellish ancient river (4,5)
I believe the answer is:
hold forth
'speak in public' is the definition.
(I've seen this before)
'close to hellish ancient river' is the wordplay.
'close to' indicates one should take the final letters (the 'close' to the word).
'ancient' becomes 'old' (similar in meaning).
'river' becomes 'forth' (Scottish river).
The last letter of 'hellish' is 'h'.
'h'+'old'+'forth'='HOLD FORTH'
(Other definitions for hold forth that I've seen before include "speak in public?" , "make a speech" , "Speak lengthily, maybe preach" , "Offer" , "Present" .)