Hear he abandoned the child and that's a crime (5)
I believe the answer is:
arson
'that's a crime' is the definition.
(I've seen this before)
'hear he abandoned the child' is the wordplay.
'abandoned' suggests deleting specific letters.
'the child' becomes 'son' (son is a kind of child).
'hear' with 'he' taken out is 'ar'.
'ar'+'son'='ARSON'
'and' is the link.
(Other definitions for arson that I've seen before include "Roans (anag)" , "Criminal pyromania" , "Incendiary crime" , "Crime of fire-setting (5)" , "Fire-raiser's crime" .)