Exit speech from Atlantean, with the odd omission (2-2)
I believe the answer is:
ta-ta
'exit speech' is the definition.
('ta-ta' might be said when leaving)
'atlantean with the odd omission' is the wordplay.
'with the odd omission' indicates alternate letters (letters in odd-numbered positions are removed).
The alternating letters of 'atlantean' are 'TA-TA'.
'from' is the link.
(Other definitions for ta-ta that I've seen before include "A Farewell" , "Bye!" , "Bye-bye" , "Adieu" , "Cheerio" .)