The plot might also be that the Earl's hug is a way to get Allen on his side. He could be trying to convince Allen that there is a different path than the one he's currently on, and this hug is the first step in that persuasion.
There could be many interpretations in such a fanfiction. Maybe the Earl has some complex motives behind hugging Allen. It could be a moment of false kindness, where he's trying to manipulate Allen in some way, perhaps by making Allen lower his guard.
You can try looking on fanfiction websites like Archive of Our Own (AO3) or FanFiction.net. There are often a lot of D Gray - Man fanfictions on these platforms. Just use the search function and type in relevant keywords like 'D Gray - Man', 'the Earl', and 'Allen' along with 'hugging' to narrow down the search results.
Perhaps the Earl discovers Allen when he is in a vulnerable state. The fanfic might explore how the Earl's presence affects Allen's development. It could also show Allen's internal struggle between the kindness the Earl might show him and his loyalty to the Black Order. It could be a story full of psychological drama as Allen tries to figure out the Earl's true intentions.
Perhaps in the fanfiction, the Earl and Allen have a moment of understanding. The hug could be a sign of a truce, even if it's temporary. It could be that they both realize they have more in common than they thought, like their connection to the Innocence in some unexpected way. And this hug is a physical manifestation of that new - found understanding. However, it's also likely that the hug is a part of a more complex emotional battle between them. The Earl might be using the hug to taunt Allen, reminding him of his own inner turmoil and the difficult path he has to walk as an Exorcist.
Isolation is also a theme. Allen is often alone in his struggles, being the host of the 14th Noah. Fanfics may depict his feelings of isolation from both the Exorcist side and the Noah side, and how he copes with being different from everyone else.
Maybe the Earl hides Allen in a parallel dimension where time moves differently. Allen has to find a way to get back to his own world while also dealing with the strange rules and creatures in this new dimension. The Earl's motives for doing this could be to keep Allen away from the Order and use his power for his own evil plans later.
There are some fanfics that focus on Allen Earl's growth in power. They might describe his training sessions in more detail, and how he overcomes his weaknesses. Maybe he discovers a new form of Innocence or a unique way to use his existing powers. These fanfics often add exciting new elements to his character development.
One plot idea could be that the Earl is hiding Allen in a parallel dimension only he can access. Allen has to figure out a way to escape while the Earl tries to convince him to join his side. Another idea is that the Earl disguises Allen as one of his Akuma, and Allen has to maintain the ruse while secretly trying to undermine the Earl's plans from within. Maybe the Earl hides Allen's Innocence in a way that Allen has to go on a quest to find it again, all while being under the Earl's watchful eye.
In this brainwashed Allen fanfic, it could also be that the brainwashing is part of a larger scheme. The Millennium Earl might be trying to use Allen as a weapon against the Black Order. Since Allen has the Innocence and his special powers, if he's under the Earl's control, he could cause a great deal of damage. The fanfic could then detail the battles that occur as a result, with the Black Order having to fight Allen while also trying to save him. They might have to find some long - lost knowledge or a special item to reverse the brainwashing process.
One possibility is that the fanfic focuses on the relationships Allen has in this new state. His friends might be confused or worried about him. For example, Lenalee could be trying to reach out to him, but Allen is pushing her away because he doesn't want to put her in danger. And the Order might be divided on how to handle Allen now that he's different. They could see him as a threat or as a potential weapon against the Millennium Earl.