The thing I just said's rubbish, husband admitted (4)
I believe the answer is:
that
'the thing i just said's' is the definition.
('that' can refer to something just mentioned)
'rubbish husband admitted' is the wordplay.
'rubbish' becomes 'tat' (low-quality items or material).
'husband' becomes 'h'.
'admitted' indicates putting letters inside.
'tat' going around 'h' is 'THAT'.
(Other definitions for that that I've seen before include "Just like --; -- is to say" , "See 5" , "Yon" , "The (one) there" , "Not 8 Ac but the other" .)