Long-term relationship with no time for a dog (7)
I believe the answer is:
mongrel
'for a dog' is the definition.
(mongrel is a kind of dog)
I cannot understand how the rest of the clue works.
Can you help me to learn more?
(Other definitions for mongrel that I've seen before include "Dog of no recognisable breed" , "of unknown variety" , "unlike well-bred setter, say" , "Hybrid" , "Non-pedigree" .)