A lot of this drama is based on historical facts. Ba Wu's real name should be Kim Cheon Seok, who was the grandson (not son!) of Kim Je Nam, the previous king's father-in-law. After Kim Je Nam was executed, Cheon Seok hid for 10 years at the Yeongwon Temple in Chiak Mountain. Just as Ba Wu.
Su Gyeong's story, however, is fiction. Gwanghae did have one daughter with Royal Consort Yun, as you can see in the Wikipedia article, her name not known to us. But the princess only got married after her father's death.