In Cicero’s time this drink becoming a problem? (6)
I believe the answer is:
hiccup
'a problem?' is the definition.
(I've seen this before)
'in cicero's time this drink' is the wordplay.
'in cicero's time this' becomes 'hic' ('this' in Latin - Cicero was Roman).
'drink' becomes 'cup' (a cup of drink).
'hic'+'cup'='HICCUP'
'becoming' acts as a link.
(Other definitions for hiccup that I've seen before include "Involuntary spasm of the diaphragm" , "Minor difficulty" , "Sudden throat sound or minor mishap" , "Reflex spasm of the diaphragm" , "Involuntary glottal closure" .)