After start of speech, catch sound of man's slow delivery (5,4)
I believe the answer is:
snail mail
'slow delivery' is the definition.
(I've seen this before)
'after start of speech catch sound of man's' is the wordplay.
'after' means one lot of letters go next to another.
'start of speech catch' becomes 'snail' (I am not sure about this - if you are sure you should give a lot more credence to this answer).
'sound of' shows a homophone (sound like).
'man' becomes 'male'.
'male' sounds like 'mail'.
'snail' next to 'mail' is 'SNAIL MAIL'.
Can you help me to learn more?
(Other definitions for snail mail that I've seen before include "Ordinary post" , "Postal system (facetious)" , "Conventional (slow) post (informal)" , "Modern jargon for letter post, mollusc-slow compared to electronic transport" , "The conventional postal service" .)