I agree with the people here, I think she was pregnant and that was the third person he killed. That would be the secret of who he killed and it explains why he can't say it because that would directly give away the third person. The only faults I have with this theory is she said she hadn't told her friends which I thought implied that this was an old secret but maybe not, and Yuichi said that he wish she hadn't told him that which would also imply it would a secret that was able to be kept. And I think the reason he would have killed her for it is because he would have had to share her, I think he saw her as his. His mother. No one else's. I mean, I love Yuichi as much as the fanatic weeb, but can you imagine him as a sibling? Yeah, exactly.
I tripped up the stairs <3