Mop up for the dog! (3)
I believe the answer is:
pom
'the dog' is the definition.
(short for Pomeranian)
'mop up' is the wordplay.
'up' is a reversal indicator (in down clue: letters go upwards).
'mop' backwards is 'POM'.
'for' is the link.
(Other definitions for pom that I've seen before include "Small dog (colloq.)" , "Australian term for Englishman" , "Little dog" , "Britisher in Oz" , "Limey in Oz" .)