Yes, there are several. Some focus on exploring how Percy's life would be different if he was born a monster. For example, in some fanfics, his relationships with other characters are completely reimagined. He might face more prejudice or have to find new ways to fit in the world of demigods and monsters.
Well, a major theme is the struggle for power. In some fanfics, being born a monster gives Percy different powers, and he has to learn how to control them. Also, friendship is often explored. Some stories show how his relationships change when he's a monster. For instance, some of his old friends might be scared of him at first, but then they learn to accept him again. This shows the complexity of friendship. And there's often a theme of redemption too. Percy may have to prove that just because he was born a monster doesn't mean he has to be 'evil'.
Obviously, Percy is the main character. There could also be his family members who might react in different ways to his being born a monster. Some might be accepting, while others could be horrified.
The theme of powerlessness is quite common in 'percy begs pjo fanfic'. Percy, who usually has some level of control in the main series, is now in a position where he has to beg. This shows that there are forces or situations that are beyond his normal ability to handle. And there's also the theme of hope, as his begging could be a last - ditch effort to find a solution to a seemingly impossible situation.
One common theme is adventure. Percy is always going on some sort of journey or quest. Another is friendship, as he relies on his friends like Annabeth and Grover. And of course, there's the theme of power, especially related to Percy's unique abilities which might be emphasized more in fanfics about his tail.
The relationship can also be shown as very loving. Poseidon might secretly watch over Percy and be proud of his son's achievements. He could send signs of his support during Percy's battles or quests. And Percy, in turn, is always eager to make his father proud and show that he is a worthy son of Poseidon.
Often, there are storylines where Percy and Poseidon have to work together to save the world. Maybe a primordial force is rising and only their combined powers of a demigod and a god can stop it. Also, some fanfics explore the family dynamic between them, like Percy trying to understand his father more deeply, and Poseidon trying to be a better father despite his godly duties.
One popular 'percy begs pjo fanfic' is 'Percy's Plea'. It really delves into Percy's character in a new and interesting way. The story has some great emotional moments where Percy is in a desperate situation and has to beg for help. It also ties in well with the overall PJO universe lore.
One popular Percy Tail PJO fanfic is 'The Lost Hero's Quest'. It takes the characters on a new adventure filled with mystery and danger. Another great one could be 'Percy and the Underworld Revisited' which delves deeper into the relationship between Percy and the Underworld. There's also 'Tail of the Titans' that has an interesting take on the battles against the Titans from Percy's perspective.