Heat and cut in pieces on your own (10)
I believe the answer is:
unattached
'on your own' is the definition.
(not attached to anyone else)
'heat and cut in pieces' is the wordplay.
'in pieces' indicates anagramming the letters.
'heat'+'and'+'cut'='heatandcut'
'heatandcut' is an anagram of 'UNATTACHED'.
(Other definitions for unattached that I've seen before include "Not married" , "Not joined" , "Not belonging" , "Single" , "independent" .)