“Can I finish it?”
Your boyfriend offers you some of his drink. It's more than half gone, and you'd like to drink the rest.
Can I finish it?
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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.