Maudlin, oddly missing European Union: goodbye! (5)
I believe the answer is:
adieu
'goodbye' is the definition.
(synonyms)
'maudlin oddly missing european union' is the wordplay.
'oddly missing' indicates alternate letters.
'european union' becomes 'eu'.
The alternate letters of 'maudlin' are 'adi'.
'adi'+'eu'='ADIEU'
(Other definitions for adieu that I've seen before include "cheers" , "12 Down" , "I'm going" , "See you later" , "Goodbye to the French" .)