I have not had a chance to watch episode 10, but I have watched way too many K-Dramas, read way too much British literature, and studied far too much European history filled with court power struggles and intrigue because I am stuck on a conspiracy theory after episode 9.

Hui-Jui’s sister-in-law stated that I-An would be king if divorce in the monarchy were possible hinting at deeper court knowledge. I-An had been closer to Yi-Rang and shielded her by escorting her to royal events in place of his frequently absent brother Crown Prince I-Hwan. I-An mentioned Princess Diana in an episode indicating his familiarity with the real-life drama of a crown prince who should have married the woman he loved instead of marrying and neglecting a member of the aristocracy. I-An spoke with Yi-Rang about her and I-Hwan’s marriage woes.

Everyone naturally assumes that Yi-Rans is I-Yoon’s birth mother, but Yi-Ran’s disdain for I-Yoon’s being “weak just like his father” hints at a birth switch where I-Yoon actually is the king’s son and I-An’s nephew but not Yi-Rang’s son. This makes sense if I-Hwan had a mistress he had gotten pregnant and wanted to abdicate the throne and divorce, but Yi-Rang blocked that option by way of a real or fake pregnancy. Her brothers at the royal hospital could have swapped babies if Yi-Rang was pregnant with a daughter or even faked a pregnancy to secure their family connection with the crown as often seen in K-Dramas. Any 3rd party DNA paternity test of a baby passed off as Yi-Rang’s that was in fact is a son fathered by the king and born to a mistress would only confirm the important royal lineage.  As part of a birth swap, I-Hwan’s mistress and their baby naturally would have “both died in childbirth“ leaving him grief stricken. Later, in fear for his life after discovering the truth and learning the depth of treachery of Yi-Rang’s family, I-Hwan decided to abdicate and renounce I-Yoon’s claim to the throne in favor of I-An to protect himself and I-Yoon. With nothing to connect I-An and the throne to Yi-Rang’s family, I-An would also be safe from their manipulation.

We now know that I-An and I-Yoon were in the next room playing with cars as Yi-Rang fought with her husband, King I-Hwan. I-An must have overheard this and learned that I-Yoon was the queen mother’s pawn but not child just before the fire and I-Hwan’s death. Raised to follow his duty to the crown and with love for his now deceased brother I-Hwan, I-An naturally sought his nephew immediately after the fire and remained close by to preserve his nephew’s place on the throne with as little influence from the queen mother’s family as possible. Although captive, this would at least protect I-Yoon from physical harm without exposing a scandal that would destroy the monarchy.

I put this drama on hold awhile back and altho I've read recaps, have not seen all of the "flashbacks". My early theory was (and the flashbacks I haven't seen might totally contradict this), if Yi-rang and Yi-hwan didn't have a child, he could divorce her because succession was critical. They were married for 8 years before Yi Yoon was born. Perhaps Yi-hwan couldn't get a woman pregnant, so she got pregnant with unknown man. Part 2 of my theory - PM is the father.

This would also explain why Yi Hwan would abdicate to his brother - he knew (or figured out) that Yi Yoon wasn't his son and didn't deserve to succeed to the throne. It never made sense to me that Yi Hwan would deliberately cut his own son (the rightful heir) out of succession simply because he was so young.

As for Queen Mother, her whole life revolves around maintaining her status as Queen Mother, so she absolutely would struggle to ensure that Yi Yoon remains King.

Your theory is a good one, especially if there are flashbacks showing that Yi-Hwan had mistresses or was in love with another woman before he married.