In English we would say, “Can I” but technically it would be better to say, “May I.” Is this encompassed all in Kann ich, or would it be better to say, “Darf ich.” My grade school English teacher would remind us of this if we’d ask to use the washroom. Pupil: “Can I use the washroom.” Teacher: “I don’t know if you can, but you may.”
German is pretty similar here: as a native speaker in both languages, I’d translate “Kann ich” into English as “Can I” - fine but not particularly polite. To me, “darf ich” emphasises the question of permission - appropriate e.g. in your school example. If you’re just trying to be polite, I would personally use the conjunctive: “Könnte ich”/“Dürfte ich” (in the conjunctive I perceive these as fairly interchangeable, but other native speakers may of course differ).