It's very possible that there is a fanfiction with this concept. Many fanfiction authors enjoy creating unique relationships between characters from different fictional realms. In such a fanfiction, the author would probably have to establish some sort of backstory for how Hermione could be King Arthur's daughter. Maybe there's some time - traveling or alternate - universe - type of magic involved. To find it, you should search for it on popular fanfiction platforms, and also look at any related tags or fandoms that might be associated with this particular crossover idea.
The relationship between Hermione and King Arthur could be very interesting. Arthur, as a noble and chivalrous king, might be initially skeptical of Hermione but soon come to admire her intelligence. Hermione, for her part, could see in Arthur a leader with a different kind of leadership than what she is used to in Dumbledore. They might start off as allies in the face of a common threat and gradually develop deeper feelings for each other as they get to know each other better.
A possible plot in Hermione and King Arthur fanfiction could involve a threat that requires the combined knowledge of both their worlds to defeat. Maybe an ancient evil has risen that can only be vanquished by using the precise spell - casting techniques of Hermione's world along with the magical artifacts and prophecies from King Arthur's world. This would force them to work together, and along the way, they could develop a deep respect and friendship for each other.
I'm afraid I can't recommend a particular fanfiction with this plot. There are so many fanfictions out there that it's difficult to keep track of all the unique concepts. You could try looking at fanfiction communities or forums related to Harry Potter and King Arthur legends. There, other fans might be able to point you in the right direction.
One reason could be for the continuation of his dynasty. In the context of Camelot, having a son would mean there is an heir to the throne. It would ensure the stability and future of his kingdom. Without a son, there could be disputes and power struggles over who will succeed him.
Well, in such a fanfiction, they could team up against a common enemy. Maybe an ancient dark force that has been awakened in both their times. Their combined knowledge of magic from different eras would be a great asset.
I'm not sure of a specific title right off the bat, but you can try searching on fanfiction platforms like Archive of Our Own. Look for tags like 'Hermione', 'Arthur', and 'pregnancy'. You might find some hidden gems there. There could be some where the pregnancy is a result of a magical accident or a new spell gone wrong, which makes for an interesting plot.
Maybe Prince Arthur is on a quest to find his true identity and he discovers that King Arthur is an important part of his past or future. They meet and King Arthur recognizes a great potential in Prince Arthur. He takes him under his wing and together they face a common enemy, like an evil sorcerer who is threatening Camelot. Along the way, Prince Arthur has to prove himself worthy to King Arthur and the knights of the Round Table.
There are many fanfictions out there, but the idea of Hermione cheating on Ron with Arthur is not in line with the original spirit of the Harry Potter series. It's likely a very unorthodox and non - canonical fan - created scenario. I would not recommend seeking out or promoting such content that goes against the positive relationships in the original story.
One possibility in Hermione and Arthur Pendragon fanfiction could be a story where Hermione uses her modern magical knowledge to help Arthur Pendragon in his time. For example, she could introduce new spells or ways of using magic that were not known in his era. This could lead to a series of adventures as they face challenges together, like defending Camelot from a new dark magic threat.