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.
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.
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.
Lavi, who was once close to Allen, might be the most conflicted. He wants to believe in his friend but the fact that the Earl was involved in hiding Allen makes him question everything. Their friendship could turn into a very complex and tense relationship with Lavi constantly watching Allen's every move.
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.
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 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.
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.
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.