Im British. This is quite a normal thing for us to say. We say both "I'll have tea" and "I'll have A tea". A bit like saying "I'll have a drink". We might also say "I'll have a coffee/beer/wine". I cant speak for everyone in Britain, but "I'll have tea" would be used more if someone's just asked what you want to drink, so you already know that a drink is being talked about. Where as "I'll have a tea" or "Can I have a tea" would be used if you were asked a less specific question like "can I get you anything?" If you were in a bar and caught the bartenders attention and said "can I have beer" it would sound weird! But "can I have a beer" would sound ok.
This is only a literal translation. If this site is going to build true fluency, it must take into account idomatic differences. In Italian, it is ok to say "un te", but in order to get the most accurate and grammatically correct translation, it must take into account the proper, not just the direct, translation.
In this case - "the tea" would be better, and "some tea" would be most proper.
"a" really just means "one". Although there is a bit of confusion as to saying "a tea", as seen in the rest of this discussion: if we use an example of, say, "a biscuit" we really mean one biscuit. They just put "un" in there to distinguish between "the biscuit"and "a biscuit".... except they're talking about tea...
Not everywhere in the world. Just to add to confusion, if you say 'We will have tea soon' in Australia, it means 'We will have dinner soon'. And I need to further clarify, because in rural Australia, in my experience, 'dinner' is the mid-day meal and 'tea' is the evening meal. If we are sitting around and feeling like a hot drink, we would say ANY of 'Would you like a tea? Would you like a coffee? Would you like tea? Would you like coffee? How about a cuppa? Or, simply Coffee? or Tea?
I think the main questions here are- is this a natural sentence in Italian (i.e. is it a usual way to express this in Italian)? and how many translations of this does sentence DL accept? As a native English speaker from New Zealand I paused before answering, as the presence or absence of the 'a' in this sentence would be very context driven for me. The DL translation is not wrong and the comments from English speakers other than American show this. However, for non native English speakers, I suggest the two most universal translations for this would be ' The girl drinks tea' (i.e. no 'a') and 'the girls drinks a cup of tea'. I think DL accepts both.