Wife's responsibility mostly to children (5)
I believe the answer is:
dutch
'wife's' is the definition.
(slang term)
'responsibility mostly to children' is the wordplay.
'responsibility' becomes 'duty' (synonyms).
'mostly' means to remove the last letter.
'to children' becomes 'ch' (genealogical abbreviation).
'duty' with its last letter removed is 'dut'.
'dut'+'ch'='DUTCH'
(Other definitions for dutch that I've seen before include "From Amsterdam, perhaps" , "mate?" , "Better half" , "From Holland" , "European - wife" .)