I remember in grade school being taught that we should use “may I” when we want permission, and “can I” when we’re talking about ability.
That rule seems to have fallen by the wayside. I remember writing some math curriculum for IBM twenty years ago, and the subject matter experts insisted we use “can” in the word problems when we writers would have used “may.” Sigh.
See below.
To add insult to injury, I frequently see “may” used, especially in writing, when the writer means “might.”
I know, linguistic change and all that, but you lose precision when you take away meanings.
Can—ability. You can do it if you try.
May—permission. You may do it if someone lets you.
Might—possibility. You might get permission if you ask nicely.
You can still get away with using the correct words, and no one will criticize you. And your writing will be better.