Learn Polish: Czy mogę dolać mleka do twojej kawy? - Can I add some milk to your coffee?

This is the Partitive Genitive, which is the focus of your current lesson. Verbs of accumulation or addition, especially those starting with the prefix do- like dolać (to add by pouring), take the Genitive case to mean 'some of' an uncountable substance. If you used the Accusative mleko, it would imply pouring the entire existing quantity of milk rather than just adding a splash.

How does the prefix do- change the base verb lać (to pour)?

The base verb lać simply means 'to pour' and is imperfective. Adding the prefix do- makes the verb perfective and specifically adds the meaning of topping something up or adding to a substance that is already there. You use dolać because the coffee is already in the cup and you are just supplementing it with milk.

Why is it do twojej kawy instead of just using the Dative case to show what the milk is being added to?

In Polish, actions that involve physically moving or transferring a substance into a container or mixture require the preposition do followed by the Genitive case to show the destination. The Dative case is mostly used for the animate beneficiary of an action (like giving something to a person), but not for the physical destination of a poured liquid.

Is Czy mogę the best way to offer someone milk in a formal setting?