Can I (do something)?

There are lots of different ways to ask for permission to do something:

Can I ___? May I ___? Would it be all right if I ___? Would you mind if I ___?

In the situation above, you don't need to be too polite. So "Can I ___?" is a good choice. Use "Can I ___?" when you want to do something that doesn't cause trouble for the other person. It's also good for asking questions of people that you're close to or on a similar social level with.

finish (some food or drink)

To "finish" a drink, a snack, or some food means to eat or drink the last bit of it.

Hey! You finished it? I wanted some more!

Can you finish that for me? I'm full.