There are two dinim in admitting to a k'nas: 1. Admitting is not regarded as a hoda'as ba'al din to be mechayev himself to pay k'nas because only eidim can force one to pay k'nas. 2. The hoda'ah is not just an inability to be mechayev, it is actually an exemption, therefore even if eidim show up later, he would not be chayev. The machlokes in the gemara about מודה בקנס ואח"כ באו עדים is a fundamental argument whether the concept of hoda'ah is that it is not powerful enough to force the payment of k'nas, but is not an exemption, or do we view it even as an exemption so that even when eidim come after to be mechayev him, he remains exempt.
R' Hamnuna (who is supported by tanaim) makes a peshara. If the hoda'ah is not mechayev him to pay even a principal payment, we assume that the entire goal of his admission is to exempt himself. That type of hoda'ah does not have the status of "hoda'ah" to be an exemption, it is just not enough to be mechayev him. Therefore, he is patur until eidim show up, but at that point he is chayev. However, when he is modeh in a way that he is mechayev himself to pay the principal through his admission, we regard that hoda'ah as a real hoda'ah that serves as an exemption to patur him even when eidim come later. Rashi (d.h. she'harei) makes it very clear that hoda'ah is only an exemption when it is done with the right intention, namely to return the money that was damaged or taken. Why does his intention make a difference? It seems that the concept behind k'nas is to prevent the crime. Therefore, when one steals and is mo'deh to the principal he is regretting the crime he committed and doesn't need any "inspiration" or push to avoid it in the future. But, if one is merely admitting because he wants to exempt himself from paying, which we assume to be the case when the admitting isn't mechayev him to pay a principal amount, he still needs the penalty as a preventative measure for the future, and therefore is going to be chayev when eidim come later.