Next to son, a lot of the time hawk is sounding moderate (4-5)
I believe the answer is:
soft-pedal
'moderate' is the definition.
(I've seen this before)
'next to son a lot of the time hawk is sounding' is the wordplay.
'next to' says to put letters next to each other.
'son' becomes 's' (genealogical abbreviation for son).
'a lot of the time' becomes 'oft' (often).
'hawk' becomes 'peddle' (both can mean to sell or trade).
'is sounding' indicates a 'sounds like' (homophone) clue.
'peddle' sounds like 'pedal'.
's' put next to 'oft' is 'soft'.
'soft'+'pedal'='SOFT-PEDAL'
(Another definition for soft-pedal that I've seen is " Tone down".)