Breathe on that woman’s cat (7)
I believe the answer is:
panther
'cat' is the definition.
(panther is a kind of cat)
'breathe on that woman's' is the wordplay.
'breathe' becomes 'pant' (panting is a kind of breathing).
'on' says to put letters next to each other.
'that woman' becomes 'her' (I've seen this before).
'pant'+'her'='PANTHER'
(Other definitions for panther that I've seen before include "See 2" , "Black leopard" , "Bagheera?" , "kind of feline" , "Then rap the large black cat" .)