Pets going head to head inside very well broken up (8)
I believe the answer is:
staccato
'broken up' is the definition.
(musical term describing short, sharp notes)
'pets going head to head inside very well' is the wordplay.
I cannot really understand how this works, but
'to' is found in the answer.
This may be the basis of clue (or it may be nonsense).
(Other definitions for staccato that I've seen before include "Cat's coat (anag) - clipped" , "not to be played legato" , "not running smoothly" , "With notes sharply separated (music)" , "With sharply-detached notes" .)