Communicate quietly with that man's reputation rising (7)
I believe the answer is:
whisper
'communicate quietly' is the definition.
(I've seen this in another clue)
'with that man's reputation rising' is the wordplay.
'with' becomes 'w' (short for with).
'that man's' becomes 'his' (belonging to a man).
'reputation rising' becomes 'per' (I can't justify this - if you can you should believe this answer much more).
'w'+'his'+'per'='WHISPER'
Can you help me to learn more?
(Other definitions for whisper that I've seen before include "Speak very quietly" , "Rumour" , "Talk in a low voice" , "Spread rumours" , "Speak in a low voice" .)