He might go to Beijing last month.
He might have gone to Beijing last month.
He could have gone to Beijing last month.
Any difference in meaning?
Expresses a possible future action. Therefore the use of last month is impossible. You must use a time expression that goes with the future, for example:
Both express a possible situation in the past.
According to context could mean:
There was a possibility for him to go to Beijing last month but he didn't go.
The difference between can/could and may/might in present-day English† is the difference between ability and possibility.
†As recently as fifty years ago may/might was also contrasted with can/could in formal usage as denoting permission rather than ability. But this distinction has been steadily declining for two hundred years, and today can/could is used freely to express permission.
So #2 means perhaps he went to Beijing last month, but we don't know. Maybe he went. Maybe he didn't.
And #3 means he was capable of making a trip to Beijing last month. It says and implies nothing about whether he did or not.