Disgrace to follow: put up with that (8)
I believe the answer is:
doghouse
'disgrace' is the definition.
(as in being 'in the doghouse')
'follow put up with that' is the wordplay.
'follow' becomes 'dog' (I've seen this before).
'put up' becomes 'house' (put up can mean to house or accommodate).
'with that' means one lot of letters go next to another.
'dog'+'house'='DOGHOUSE'
'to' acts as a link.
(Other definitions for doghouse that I've seen before include "Place for one in disfavour" , "where one is in disgrace?" , "A disgraceful thing" , "for a boxer to get into?" , "Location of disfavour - kennel" .)