Bill turned up with ambassador to have a swim (5)
I believe the answer is:
bathe
'a swim' is the definition.
(bathe is a kind of swim)
'bill turned up with ambassador' is the wordplay.
'bill' becomes 'tab' (tab is a kind of bill**).
'turned up' says the letters should be written in reverse.
'with' says to put letters next to each other.
'ambassador' becomes 'he' (His or Her Excellency).
'tab' in reverse letter order is 'bat'.
'bat'+'he'='BATHE'
'to have' is the link.
(Other definitions for bathe that I've seen before include "Wash; swim" , "Immerse in liquid for wash" , "Steep" , "Wash in water" , "Wash, soak" .)