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.
Fanfiction.net could also be a good place to look. However, the organization of the stories there might not be as neat as on AO3. But still, it has a vast collection. Go to the site, use the search function and type in the relevant keywords to find the fanfic you're interested in.
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.
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.
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.
For Allen, you can focus on his internal struggle. Since he's being hidden by the Earl, he might be conflicted between his hatred for the Earl and the need to survive. Show his determination to escape and continue his mission. For the Earl, explore his motives for hiding Allen. Is it just to toy with the Black Order, or does he have a deeper, more sinister plan? Develop his character by showing his different sides, like his moments of calm calculation and his fits of maniacal laughter.
One idea could be a story where Mana and the Earl have to team up against a common enemy that threatens their existence in ways they haven't faced before. This would force them to put aside their differences and use their combined powers.
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.
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 some fanfics, Allen might be depicted as dead, but it really depends on the specific plot and the author's creativity. Fanfic writers often take liberties with the original story, so there could be various interpretations.