Tear after mongrel that’s in fashion (7)
I believe the answer is:
current
'in fashion' is the definition.
(thesaurus)
'tear after mongrel' is the wordplay.
'tear' becomes 'rent' (rent is a kind of tear).
'after' is a charade indicator (letters next to each other).
'mongrel' becomes 'cur' ('cur' can be a synonym of 'mongrel').
'rent' after 'cur' is 'CURRENT'.
'that's' acts as a link.
(Other definitions for current that I've seen before include "Flow of electricity" , "Contemporary" , "Popular" , "Electrical flow" , "Flow, of a river say" .)