I came with him, a drunk, around British resort city (5,5)

'resort city' is the definition.

'i came with him a drunk around british' is the wordplay.
'with' says to put letters next to each other.
'drunk' indicates anagramming the letters.
'around' means one lot of letters goes inside another.
'british' becomes 'b' (abbreviation e.g. in 'BBC').
'i'+'came'+'him'+'a'='icamehima'
'icamehima' with letters rearranged gives 'miamieach'.
'miamieach' enclosing 'b' is 'MIAMI BEACH'.