Review Modal verbs by function here
We use "Might" to express a slight or weak possibility.
Perhaps we shouldn't eat at the curry house, your guests might not like spicy food
We might be able to afford a trip to the beach if I get paid in time.
We use "Might" + have + participle to express such possibilities that did not come to pass in the past:
We might have been able to afford a holiday but I didn't get paid.
"Might" can be used in a very formal way to make enquiries or ask permission. It is much more common to use can, could or may.
Might I ask where you bought that shirt?
Might I enquire as to his whereabouts?
We can use "Might" to give advice or make suggestions sound more polite, especially when used with verbs such as "like", "prefer" or "want":
You might like to reconsider your course of action
He might want to try our home delivery service
We use "Might" + have + participle to express criticism or disapproval of an action
You might have told us you were not coming for Christmas
He might have tried to explain how he felt rather than simply leaving
When we report speech, some verbs are changed. "May" is reported as "Might"
"She may have had a change of heart" siad Brian.
Brian said she might have had a change of heart,