Owns article one sort of dog ate, so getting another dog (5,4)
I believe the answer is:
lhasa apso
'another dog' is the definition.
'Lhasa apso' can be an answer for 'dog' (Lhasa apso is an example). I'm not sure about the 'another' bit.
'owns article one sort of dog ate so getting' is the wordplay.
'owns' becomes 'has' ('have' can be a synonym of 'own').
'article' becomes 'a' (the indefinite article in English).
'one sort of dog ate' becomes 'lap' (I can't justify this - if you can you should give a lot more credence to this answer).
'getting' is an insertion indicator.
'lap'+'so'='lapso'
'has'+'a'='hasa'
'hasa' going inside 'lapso' is 'LHASA APSO'.
(Other definitions for lhasa apso that I've seen before include "Tibetan breed of dog - a salsa hop (anag)" , "small dog" , "Shown [AT CRUFTS]" .)