Annoyed with one child encountered in the capital of Wisconsin (7)
I believe the answer is:
madison
'the capital of wisconsin' is the definition.
(synonyms)
'annoyed with one child' is the wordplay.
'annoyed' becomes 'mad' (mad can mean annoyed or angered).
'with' is a charade indicator (letters next to each other).
'one' becomes 'i' (Roman numeral).
'child' becomes 'son' (male child).
'mad'+'i'+'son'='MADISON'
'encountered in' is the link.
(Other definitions for madison that I've seen before include "James, fourth US president" , "New York Avenue for advertising people" , "US president, county and square" , "4th president of the US - capital of Wisconsin" , "US president - New York square with gardens" .)