So for those who want a summary of why no one is happy about the new emperor:
The previous king wrote a letter to Alfort, the name of his little brother, leaving the land to him as a way of trying to chase away his demons. The one who suggested it is Farna, who only suggested it to get him to stop fucking her. Her son is also named Alfort (bad naming there, parent) who she hasn't raised well as a ruler and instead wants to take him away from any ruler succession and give him land where he can live in peace.
Issue is, said coping letter is found by the prime minister who decides to interpret it as a last will instead. This upsets Farna because this is exactly what she didn't want. Issue is her son, being naive as he is, asked his mother if he is safer like that, essentially making it an acceptance of the throne in the prime minister's eyes. Farna is also a mistress at best so even if she is to object herself, she has no power or say in the matter.
Tillia is naturally not happy with this situation because she was getting ready to be the successor. However according to Farna the prime minister seems to have bad blood with Tillia's mother, thus has bad blood with Tillia. It also seems Tillia is aware of some people in power abusing their power for self interest and wants that changed, which i assume the prime minster doesn't want.
In summary, the only one really getting what they want is the prime minister and perhaps those who work with him. They get Tillia out of the way, they have a weak person as a puppet ruler and Farna is left in a situation where her worst case scenario comes true and there is no peace for her.