Twenty runs — noted down by him? (6)
I believe the answer is:
scorer
'noted down by him?' is the definition.
The answer is a person as well as being a singular noun. This is suggested by the definition.
'twenty runs' is the wordplay.
'twenty' becomes 'score' ('score' can be a synonym of 'twenty').
'runs' becomes 'r' (cricket abbreviation).
'score'+'r'='SCORER'
(Other definitions for scorer that I've seen before include "He scratches" , "Footballer having netted" , "One keeping count of points earned" , "Sir, not one without heart as a" , "One gaining points" .)