Such a nuisance writing an advertising slogan for a cigarette! (4,1,3)

'such a nuisance writing' is the definition.
I don't know anything about this answer so I can't tell whether it can be defined by this definition.

'advertising slogan for a cigarette' is the wordplay.
'advertising slogan' becomes 'what' (I am not sure about this - if you are sure you should believe this answer much more).
'for' is a charade indicator (letters next to each other).
'cigarette' becomes 'fag' (synonyms).
'what'+'a'+'fag'='WHAT A FAG'

'an' acts as a link.

Can you help me to learn more?