Well up about care of cat (7)
I believe the answer is:
scourge
'cat' is the definition.
(I've seen this before)
'well up about care of' is the wordplay.
'well up' becomes 'surge' (I've seen this before).
'about' indicates putting letters inside.
'care of' becomes 'co' (c/o can mean 'care of' when addressing mail).
'surge' enclosing 'co' is 'SCOURGE'.
(Other definitions for scourge that I've seen before include "Cause great suffering to (someone); whip" , "Curse; whip" , "Person/thing causing great suffering; whip" , "Whip; agent of suffering" , "Whip used to inflict punishment" .)