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.