Yes, there might be. The Hunger Games fandom is quite large and diverse, and fans often come up with creative and sometimes unexpected scenarios in fanfiction. This particular concept, while a departure from the original story, could be explored by some fanfic writers who like to play with the relationships and family trees in the Hunger Games universe.
One interesting Katniss and Snow fanfiction could be one where Snow has a change of heart before the Hunger Games start. Maybe he realizes the brutality of the system he's created and tries to secretly help Katniss. Another could be a story set in an alternate universe where Katniss and Snow are forced to work together to save Panem from a greater threat.
Well, Katniss is the protagonist who has been through so much in the Hunger Games world. She is a symbol of resistance. Snow is the antagonist, a powerful and often cruel leader. In this fanfiction, they are likely to be central to any storylines, with their personalities and the dynamic between them being explored in different ways. For instance, their interactions could range from intense hatred to an unexpected understanding depending on how the fanfiction is written.
You can try websites like Archive of Our Own (AO3). It has a large collection of fanfictions from various fandoms, including The Hunger Games. Just search for 'Katniss and Snow' and you'll likely find many different stories to choose from.
Well, perhaps in this fanfiction, Snow had a secret child long ago who was Katniss's parent. Katniss grows up in District 12 unaware of her relation to Snow. When she becomes the Mockingjay and starts to dig deeper into the Capitol's secrets, she uncovers the truth. This could lead to a complex plot where the rebels are suspicious of her loyalty, and she has to prove herself all over again while also dealing with her own internal conflict about her new - found family connection.
One interesting plot could be a scenario where Katniss and Snow are forced to work together in a post - Hunger Games world. Maybe they have to deal with a new threat that endangers both their interests.
In some fanfictions, Katniss might be depicted as pregnant. Fanfiction allows for a wide range of creative interpretations of the characters. However, in the original 'Hunger Games' series, there is no such plot. So it depends on the specific fanfiction you are reading.
Gale might also attempt to rescue Katniss. He's her friend from District 12 and is quite skilled in his own right. He would be furious to hear that Cato has kidnapped Katniss and would likely gather resources and allies to try and save her.
Her character could develop a new level of determination. Before, she was fighting for her family and district. Now, with a baby on the way, she has an even greater reason to survive. She might become more calculating, using her wits to avoid dangerous situations rather than charging in head - first. She would also probably show more vulnerability, as the pregnancy adds an extra layer of stress and worry.
Katniss would likely become more protective. Her natural instinct to survive would be amplified by the need to protect her unborn child. This could make her more cautious in her actions in the arena, but also more fierce when directly threatened.