Como dizer "Um copo com gelo, por favor"

Tenho dúvidas em relação ao uso das palavras glass e cup. Caso eu esteja em um fast food, os copos são descartáveis. Na hora de pedir um copo de gelo (em separado das bebidas) eu devo dizer: Can I have a glass with ice?
Quando usar cup ?

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Could you bring me a plastic cup with (some) ice cubes?
(meaning "with ice cubes in it").

Meaning styrofoam cup - from your explanation those are disposable cups so plastic would mean those styrofoam cups.
There are plastic cups, made of hard plastic, but I am assuming that the place offers the styrofoam/disposable one.

"Can I have" to me, has a certain ring of drink or food to eat: can I have a coffee? Can I have some water? Etc.

If there are the two kinds: glasses and styrofoam cup, then you could say "can I have a (separate) glass with ice cubes in it?" OR (with the disposable cup example) perhaps you should say "one of those disposable cups..." or you would point it out... There are several likely scenarios.

Let´s wait for more advanced learners to chime in, or natives, teachers etc. I must admit that I have not gotten myself in such a situation yet, nor I have see it in movies or read in novels, etc.
This is my educated guess, then.
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Customer: I'd like ice in my juice.
Waiter / Waitress: Sure!

Customer: May I have some ice-cold beer?
Waiter / Waitress: Yes, sir / ma'm!.

Bartender / Server: Good evening. What can I get for you?
Customer 1: Good evening. May I have a glass of cold water?
Customer 2: Are you joking, right?
Customer 1: But I'm dying of thirst.

Customer: Can I have a pint of stout and some ice cubes please?
Waiter / Waitress: Of course!
2 25 393
Flight attendant: Would you like something to drink, sir?
Passenger: I have a scratchy throat. Could you put just one ice cube in my juice?
Flight attendant: Which flavor do you prefer? Orange, mango or strawberry?
Passenger: Mango.