Child of hairdresser runs away (4)
I believe the answer is:
babe
'child' is the definition.
(babe is a kind of child)
'hairdresser runs away' is the wordplay.
'hairdresser' becomes 'barber' (barber is a kind of hairdresser).
'runs' becomes 'rr' (two abbreviations for run).
'away' suggests deleting specific letters.
'barber' with 'rr' taken away is 'BABE'.
'of' acts as a link.
(Other definitions for babe that I've seen before include "Term of endearment" , "One of the little people" , "Ruth possibly" , "Ruth, for example" , "Ruth, say" .)