Is "might could" a correct construct?

No, it is not.

This construct would not have a clear meaning, as might is a form of "may" and expresses a possibility - but "could" has a very similar meaning.

What you can say, however is

"might be able to" - here the "could" is replaced by a more specific "be able to" which specifies that it's someone's ability or permission, rather than possibility - then the phrase is no longer redundant with two words meaning almost the same.