Suffer at home, remedy falling short (5)
I believe the answer is:
incur
'suffer' is the definition.
(I've seen this before)
'home remedy falling short' is the wordplay.
'home' becomes 'in' ('he's home' can mean 'he's in').
'remedy' becomes 'cure' ('cure' can be a synonym of 'remedy').
'falling short' means to remove the last letter.
'cure' with its final letter taken away is 'cur'.
'in'+'cur'='INCUR'
'at' is the link.
(Other definitions for incur that I've seen before include "Bring on oneself, say costs" , "Become liable to (eg, a penalty)" , "Bring something unpleasant on oneself (5)" , "Take on (obligation)" , "Fall into - invite" .)