This has less to do with crown prince, but Eira herself.
Eira is supposed to be the main character of the prophecy, which is saying that the woman possessing the holy mark will be the empress. Everyone assumes that since she didn't manifest it while with the crown prince she is not the chosen one from the prophecy.
But we know she is the one, since she actually has the mark. Just not with the crown prince. She likely will be the empress then. And her partner is obviously River, since no one else including that blonde duke managed to manifest her mark.
But she has to have a legitimate reason to become the empress besides the mark, which makes it likely that River has royal blood. His adoptive father, the count, clearly knows something, and that priest realized something as well.
It feels like the crown prince is the empress's illegitimate child while River is the emperor's illegitimate child. And Eira unfortunately became a sort of DNA tester with her blessing, which is why it didn't work on crown prince.