The deposed crown prince who the current king wants located and probably killed because he wants to stay in power.…
I don't think they have explained it properly in the show except for showing the family on the run. Probably what you said about the prince having a strong claim to the throne is true. Maybe they weren't father and son but step-brothers which is why he is now desperate for his own son.
It is a bit weird if they go in that direction. As per the synopsis, she's the daughter of his love. Anyway, feel free to spoil for me when you watch episode 10.