Speech made when a rotten rat’s about to leave (5)
I believe the answer is:
argot
'speech' is the definition.
(argot is a kind of speech)
'rotten rat's about to leave' is the wordplay.
'rotten' indicates anagramming the letters.
'about' is an insertion indicator.
'to leave' becomes 'go' (**).
'rat' is an anagram of 'art'.
'art' placed around 'go' is 'ARGOT'.
'made when a' is the link.
This may not be correct. Some or all of it may be part of another bit of the clue.
Can you help me to learn more?
(Other definitions for argot that I've seen before include "Slang for a groat (5)" , "Talk of thieves" , "Kind of slang" , "Cant" , "Slang of a particular group, originally thieves" .)