Clean up old man with grim haircut (9)
I believe the answer is:
pompadour
'haircut' is the definition.
(I know that pompadour is a haircut)
'clean up old man with grim' is the wordplay.
'clean' becomes 'mop' (mopping is a kind of cleaning).
'up' says the letters should be written backwards (in down clue: letters go upwards).
'old man' becomes 'pa' (both can mean someone's father).
'with' says to put letters next to each other.
'grim' becomes 'dour' (synonyms).
'mop' in reverse letter order is 'pom'.
'pom'+'pa'+'dour'='POMPADOUR'
(Other definitions for pompadour that I've seen before include "18th century women's hairstyle" , "18th century hairstyle employing a pad at the front to add bulk" , "Lady" , "High Barnet" , "Extravagant quiff" .)