We can use 'may' to ask for permission. However this is rather formal and not used very often in modern spoken English
We use 'may' to suggest something is possible
We use 'might' to suggest a small possibility of something. Often we read that 'might' suggests a smaller possibility that 'may', there is in fact little difference and 'might is more usual than 'may' in spoken English.
For the past, we use 'might have'.