Would Ex spouse have rights over common law partner?

Would Ex spouse have rights over common law partner?
Yes, after breaking up with a common-law partner there are very little rights, unless there is children involved.

A bit more:

That depends on what sort of rights you mean, as well as in what state they reside. An ex spouse would have no rights over a common law partner, except in the case of children the ex spouse had with the common law partner. Other than that, there are no rights the ex spouse would have over the common law partner.

If you were married and are now divorced, and your ex spouse is now living with a common law partner, you have no rights over that common law partner (excluding their children).

More detail would make it much easier to provide a more accurate answer, which you can post in the discussion section of the Q&A.