They have the same father but different mothers. Xu Feng is the Empress' son.
From what I remember, Run Yu's mother is dead because she is a fish sprite and sprites have a shorter lifespan than those of immortals. The Emperor became attracted to her because she reminded him of the Flower Deity, Jinmi's mother. However, Run Yu hates his father because the latter prevented his mother from becoming the Queen of a sea king, and it's not like the Emperor protected her when she was alive.
This is what I know about the setting of Habaek's Bride. The gods in the story derive from Chinese mythology and I guess the myth of Habaek shows us about the relationship between Korea and China in pre-Goguryeo Dynasty since Yu-hwa, one of Habaek and Soah's daughters, is the mother of Jumong, the founder of Goguryeo.
However, we won't see much of the fascinating mythological and historical premise as the drama adaptation would not be a sageuk.
However, we won't see much of the fascinating mythological and historical premise as the drama adaptation would not be a sageuk.