@Ox @igel I think it's much more simple:
might - accidental/uncertain
may - deliberate/certain
Be careful you might (accidentally) slip.
May I (deliberatly) help you?
You certainly may do that.
I might do that, but I'm not certain.
Maybe a bit subtle, but not that hard to pick up if you don't play the language matching game.
Now, may vs can, that's the stuff. Can is hellbent to get rid of may, but there are still a few places where may finds refuge, like... star wars, I guess...
Be careful you maybe slip.
Can I help you?