Caught Igor going off with the dog (5)
I believe the answer is:
corgi
'the dog' is the definition.
(corgi is a kind of dog)
'caught igor going off' is the wordplay.
'caught' becomes 'c' (cricket abbreviation).
'going off' indicates anagramming the letters (letters go or move around).
'igor' is an anagram of 'orgi'.
'c'+'orgi'='CORGI'
'with' is the link.
(Other definitions for corgi that I've seen before include "Cardigan, possibly" , "pet in 9 [CARDIFF]?" , "Pre-2018 royal dog" , "Dog at Buckingham Palace" , "man's best friend" .)