I understood that "might" represents a weaker possibility and "may" represents a stronger possibility. But I can't understand when I have to use "may not" and "might not". What's the difference between them?
Firstly lets get some perspective and correct your misconception. In some instances the two are directly inter changeable like with an introduction or a weak possibility. In others, like General Truths, they are not. In other situations, Permissions, Suggestions and the suggestion of a possibility might is the more polite form of may.
Hence in most situations if you are unsure of which to use I would err to the use of might over that of may
May maybe used as a possibility or a general truth.
We use may to refer to a weak possibility in the present and future:
The economy may go up or down in the next year. (I think both are possible, the economy going up or the economy going down. I am not making either one a strong possibility.)
I think I may go to the doctor today and try to get some antibiotics. (I am not very sure yet if I will go to the doctor.)
We use may in formal writing, especially academic English, to describe things which the speaker thinks are generally true or possible. In this case, it is a more formal equivalent of can.
Might may be used as the past simple of the verb may, also as a possibility, permission, suggestion or as an introduction. It can also be used for what someone should do to be pleasant, correct, polite, etc, this context is usually associated with what someone would say when the are angry.
might modal verb (MAY)
past simple of the verb may, used especially when reporting what someone has said, thought, asked, etc.: I brought him some sandwiches because I thought he might be hungry. Very politely the little boy asked if he might have another piece of cake (= he said "May I have another piece of cake, please?").
might modal verb (POSSIBILITY)
Driving so fast, he might have had a nasty accident (= it could have happened but it did not). The rain might have stopped by now.
might modal verb (PERMISSION)
Might I ask a question? I wonder if I might have a quick look at your newspaper?
might modal verb (SUGGESTION)
You might try a little more basil in the sauce next time. I thought you might want to join me for dinner.
might modal verb (INTRODUCE)
used to introduce a statement that is very different from the statement you really want to make, in order to compare the two: The amount you save might be small, but it's still worth doing.
might modal verb (SHOULD)
You might at least try to look like you're enjoying yourself!
"I asked my boss over for dinner tonight." "Well, you might have asked me first!"
See this New York Times article "May, Might, Muddle"
It's my contention that "X may not have happened" means that we don't know whether or not it did, while "X might not have happened if..." means that it did happen but could have been avoided. Newspaper articles often say something like "If it had not been for [something] he may not have died" when we know that the unfortunate person did die - but I maintain that that is incorrect.
Do you have any authority for claiming a difference in meaning between “may” and “might”?
I have never heard of such a difference. Consequently, I looked at Merriam-Webster, which makes no such distinction.
“Might” is the past tense of “may.” It is not usual in English for tense to alter fundamental meaning.
In short, I believe you are trying to cope with a difficulty that does not exist.