"Εγώ θέλω να τελειώσω το τσάι μου."
Μετάφραση:I want to finish my glass of tea.
Only a very rare English speaker in the UK. would say my glass of tea, they don't generally drink ice tea. My tea or my cup of tea is the usual expression.
So, it should be also accepted both :
I want to finish my tea
I want to finish my cup of tea
especially since the Greek sentence is not talking about ice tea.