Son matured by eating that woman in Bow's cheese (4,5)
I believe the answer is:
sage derby
'cheese' is the definition.
(I've seen this before)
'son matured by eating that woman in bow's' is the wordplay.
'son' becomes 's' (genealogical abbreviation for son).
'matured' becomes 'aged' ('age' can be a synonym of 'mature').
'eating' indicates putting letters inside.
'that woman in bow' becomes ''er' ('her' - 'her' refers to a woman - said with a cockney accent, dropping the 'H').
'aged'+'by'='agedby'
'agedby' enclosing 'er' is 'agederby'.
's'+'agederby'='SAGE DERBY'
(Other definitions for sage derby that I've seen before include "Cheese made with herb" , "Food" , "Green-veined cheese" , "Greenish cheese" .)