Cocktail of lime – men add extra designation (which each other across answer has here) (6,4)
I believe the answer is:
middle name
'extra designation which each other across answer has here' is the definition.
I know nothing about this answer so I cannot judge whether this works.
'cocktail of lime men add' is the wordplay.
'cocktail of' is an anagram indicator (a mixture of the letters).
'lime'+'men'+'add'='limemenadd'
'limemenadd' is an anagram of 'MIDDLE NAME'.
Can you help me to learn more?
(Other definitions for middle name that I've seen before include "Richard Nixon's was Milhous" , "Part of how someone is known" .)