Finish with my boy outside but dispatch after he’s left (4,2)

'dispatch after he's left' is the definition.
The definition suggests a singular noun but the answer is not.

'finish with my boy outside' is the wordplay.
'finish with' becomes 'end' (synonyms. I am not sure about the 'with' bit.).
'my boy' becomes 'son' (I've seen this in another clue).
'outside' indicates putting letters inside (some letters go outside others).
'end' placed into 'son' is 'SEND ON'.

'but' acts as a link.

Can you help me to learn more?