Tot upset sick, having swallowed hot drink in theatre (5,4)
I believe the answer is:
music hall
'theatre' is the definition.
(music hall is a kind of theatre)
'tot upset sick having swallowed hot drink' is the wordplay.
'tot' becomes 'sum' (synonyms).
'upset' says the letters should be written backwards.
'sick' becomes 'ill' (similar in meaning).
'having swallowed' is an insertion indicator.
'hot drink' becomes 'cha' (cha can mean tea).
'sum' reversed gives 'mus'.
'ill' going around 'cha' is 'ichall'.
'mus'+'ichall'='MUSIC-HALL'
'in' is the link.
(Other definitions for music hall that I've seen before include "Late 19th century entertainment, and the theatre it took place in" , "theatre of a sort" , "Mainly C 19 variety entertainment" , "Vaudeville" , "A Palace of Varieties" .)