A mild UN broadcast that gets half-cut? (7)
I believe the answer is:
maudlin
'half-cut?' is the definition.
(maudlin can mean tearfully drunk)
'a mild un broadcast' is the wordplay.
'broadcast' is an anagram indicator.
'a'+'mild'+'un'='amildun'
'amildun' with letters rearranged gives 'MAUDLIN'.
'that gets' is the link.
(Other definitions for maudlin that I've seen before include "lovey-dovey" , "Soppy, lachrymose" , "Weakly sentimental" , "Mushy and emotional" , "Foolishly tearful" .)