The words “may” and “might” can be pretty confusing. When should you use one vs. the other? Consider a sentence like “We may go to the game, depending on the weather,” which is hard to place as either more or less correct than the sentence “We might go to the game, depending on the weather.” Garner tells us that the words “occupy different places on a continuum of possibility,” with “may” expressing likelihood and “might” expressing “a stronger sense of doubt.”
So our word choice in the sentence pair above actually depends on the weather! If the weather has been gray but seems to be clearing up with a good chance of making for a good game night, then you “may” go to the game. If dark clouds are rolling in, then you “might” go.
When using one of these conditional words with a negating sentence, it’s generally going to be best to go with “might” because “may not” can be confused with “cannot,” which has a slightly different meaning.
Understanding that the two words exist along a continuum of possibility makes the little nursery rhyme a bit more poignant, for the speaker seems at first in less doubt and then in more as to the likelihood of her wish’s being granted, lending to that third line a growing sense of doubt, pathos, and urgency.