Possible spoilers:  Are Ba Wu, his father and grandfather historical figures in Joseon history?  I looked up  King Gwang-hae in Wikepedia. ", Gwang-hae tried to bring officials from various political and regional background to his court, but his plan was interrupted by Greater Northerners including Lee Icheom and Chung In-hong. Then, Greater Northerners began to take members of other political factions out of the government, especially Lesser Northerners. In 1613, the Greater Northerners moved against Prince Yeong-chang; his grandfather Kim Jenam was found guilty of treason and executed, while Yeong-chang was sent into exile, where he too was executed. "    Lee Icheom is in the drama as the king's prime enemy. . Ba Wu's father has the same name Kim Jenam who was executed.  So does that mean Cha Dol (or Ba Wu)is Prince Yeong-chang?   Do we know yet who Cha Dol's real mother is?

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.