Most writers use may and might interchangeably:
I may go to the library to work on my term paper.
I might go to the library to work on my term paper.
Is there a difference? There is, but it’s slight. May suggests a possibility that an action will occur, while might suggests a slightly smaller possibility. So if I say that I may go to the library, there’s a reasonably good chance that it’s on my agenda. But if I say that I might go, the odds that I will aren’t quite as good.
The distinction between the two is sufficiently fine that it’s not something writers need to obsess about. However, when referring to something in the past, the rules get tighter. The past tense of may is might.
She might have left a message on my voice mail. (Not she may have)
From time to time, writers also struggle with the difference between may and can. The difference here is more pronounced. May expresses permission, while can expresses ability. Moms everywhere are notorious for emphasizing this particular grammatical difference.
Question: Mom, can I paint my bedroom walls black?
Answer: I’m sure you can, but you may not.
In informal speech (including dialogue in fiction), we have slipped into using can when may would be more appropriate. In truth, strict adherence to the difference between the two can seem a little prissy at times. Still, it’s a valid distinction that writers should strive to apply when it makes sense.