Cat lady’s in pursuit of desire (7)
I believe the answer is:
panther
'cat' is the definition.
(panther is a kind of cat)
'lady's in pursuit of desire' is the wordplay.
'lady' becomes 'her' (I've seen this before).
'in pursuit of' is a charade indicator (letters next to each other).
'desire' becomes 'pant' (to pant for something is to desire it).
'her' after 'pant' is 'PANTHER'.
(Other definitions for panther that I've seen before include "Black leopard" , "Bagheera?" , "See 2" , "animal!" , "kind of feline" .)