Honeslty don't try and make sense of the situation, you'll only give yourself a sore head. The books are great for world building but jesus fucking christ this part of the story annoys me. Bear in mind the big reveal happens in the very last book so this reveal will be the end of the TV show.
Spoilers ahead! SPOILERS AHEAD.
So Emhyr was son of the king of Nilfgaard who was conquered by "The Usurper" when Emhyr was 13 years old. The Usurper then curses Emhyr to look like a hedgehog, as Emhyr means Hudgehog in Nilfgaardian. (Emhyr means "Leader" in Welsh).
Emhyr escapes and takes to calling himself Duny, The Urcheon of Erlenwald. At some point he hears a prophecy that the son of the Elder blood will rule the world. Quite coincidental then that he'd run into the King of Cintra in mortal danger who happens to be married to Queen Calanthe who is of the Elder Blood. Emhyr saves the Kings life and claims the law of surprise.
Fast forward 15 years, the King of Cintra is dead, he had a daughter called Pavetta before this happened though, and a now ~30 year old Emhyr starts banging the 14 year old Pavetta, his child of surprise.
At Pavettas 15th birthday she's supposed to be married off, her mother Calanthe has arranged, I'm secret, for her suitor to be Crach An Crait, future ruler of Skellige (imagine Viking Age Ireland.) Hedgehog boy shows up to this affair and claims his child of surprise goth ensues (we saw this in the show)
Anyway bit shit writing there already but it gets far far worse.
So Emhyr hasn't let anyone know who he really is yet, even though he will be king of the worlds most powerful nation after Calanthe dies and is very well protected. Its very understandable he wants his homeland back and wants to kill the Usurper and his ilk. Who wouldn't want to kill the man who slaughtered your family and cursed you into looking like a hedgehog. So he asks the most powerful ruler in the world, his mother in law, for help? Does he fuck. Instead he allies with Vilgefortz a very powerful mage (sword replenishing guy in the last episode) and arranges to disappear so that he can reclaim his heritage....
It makes as much sense to you as to me btw. Remember Emhyr is basically a nobody at this point, Nilfgaard is a shit hole, whereas his alias Duny is heir to the throne of Cintra!
Anyway he arranges with Veligefortz to make it seem like Emhyr, Ciri, and Pavetta all die in a shipwreck, Pavetta finds out about this plan and smuggles Ciri off board gets into a fight with Emhyr and gets pushed overboard and drowns. So Pavetta is now dead but that's okay because Emhyr claims to have never loved her....
Anyway Emhyr somehow takes back his rightful throne ends up becoming "The White Flame Dancing on the Graves of his Enemies" and greatly improves Nilfgaard and then a few years later wages war to take Cintra where his daughter still lives. Now most of this can be explained by Vilgefortz being extremly powerful and setting it all up but.... Vilgefortz goes away let's say and Emhyr then decides he has to create the son of the Elder Blood and decides he wants to marry Ciri, his daughter, which he does, and then impregnate her which he decides against and the second to last minute.