Mmmm got it. The Count was in love with the MC's mom, somehow MCs mom got pregnant with the Duke's child (I don't understand how since in the synopsis it's said the Duke is not aware either), probably the mom died/ because otherwise, I don't understand how she would abandon her daughter/ and so when the Count found about everything, he wanted to keep the child of that relationship under his power, abusing our MC nonstop, and using her in the future to his advantage.
If MC's daddy is the Duke (the more powerful noble aside from the Emperor), I highly think her mom was also a relevant figure in the social circle or had some important attribute inside her, so maybe part of the Count's attitude comes from his dejected, twisted, and disgusting feelings; but the other part comes from taking advantage of the child's identity. I'm just guessing, but since he said he might let her free at the age of 18, I think he was going to reveal her identity to then sell her to someone or to do something that would bring him benefits.
I'm leading to that theory, because it seems a bit dumb and wasteful to just sustain a child till the age of 18 to then kill her. I also think that probably the Count's daughter was the one who killed our MC by burning her, but this is just my wild imagination going on, lol. Anyway, seeing the Count's attitude, I can't stop thinking that there might be a significant secret behind MC's birth that could affect him greatly or would impact significantly the distribution of power, which is why he's acting like that.
Edit: It might even be something between the EmperorxCount since it's obvious that the Emperor is against the Duke and is super aware of his immense power and relevance. But it's too soon to tell.
And yeah, daddy is the Duke (below the Emperor), but he's the owner of the holy sword, which is the representant/ head/ king of the holly "city" (idk how to call it, but the territory where the union of the people who're under the god of this story relies upon).