Lording it, having fun with stray dog (7,3)

'lording it' is the definition.
I don't know anything about this answer so I can't tell whether this works.

'fun with stray dog' is the wordplay.
I cannot quite see how this works, but
'fun' could be 'play' (synonyms) and 'play' is found within the answer.
an anagram of 'dog' is 'god' which is located in the answer.
This explanation may well be incorrect...

'having' is the link.

Can you help me to learn more?