A sentence

I've drunk my cup of tea. Can you give me something else? Correct?

A sentence

I've drunk my cup of tea can you give me else? correct?

I've drunk my cup of tea can you give me more please? correct?

You could also write

I've drunk my cup of tea could  you give me some more  please?

or

I've finished my cup of tea could you give me some more please? 

I think that you would see the last sentence more commonly. However note that except for the word else, any English speaker would understand your request. In spoken conversation, an English speaker would ignore the error of else. They would understand your meaning.