Delay including half of underpants in washing (10)
I believe the answer is:
laundering
'washing' is the definition.
(I know that laundering is a type of washing)
'delay including half of underpants in' is the wordplay.
'delay' becomes 'lag' (lag is a kind of delay).
'including' is an insertion indicator.
'half of' indicates taking half.
'underpants' cut in half is 'under'.
'under'+'in'='underin'
'lag' enclosing 'underin' is 'LAUNDERING'.
(Other definitions for laundering that I've seen before include "when going after money - it gives offence" , "Doing the washing" .)