There could be. Try searching on different fanfic websites. Some might be hidden gems. It really depends on how popular the Adam x Lute pairing is among the Hazbin Hotel fanfic community. If it's not a very common pairing, it might take more effort to find really good ones.
Common themes may include the exploration of their personalities. Since they are characters from Hazbin Hotel, their unique traits are likely to be delved into. Also, themes of loyalty and trust could be present, as their relationship develops. They might face challenges together that test their bond, which is also a frequently seen theme.
Charlie is often paired with Lute. Since she is the owner of the Hazbin Hotel and very passionate about redemption, she might see Lute's music as a tool for helping the sinners, and they could work together on different projects.
A plot could also revolve around Lute using his music to try and rehabilitate some of the more difficult sinners in the hotel. His music has a special power that can soothe or inspire change, and he tries to use it to make the hotel a better place for everyone.
Redemption is also a frequent theme. Given the setting of Hazbin Hotel, the idea of Adam finding redemption through his relationship with the reader is quite popular. It could involve him learning about love and acceptance in a new way.
Another option is to check out Fanfiction.net. Sometimes, there are some great Hazbin Hotel Adam x Reader fanfictions hidden among the many works on the site. It might take a bit more searching compared to AO3, but it's still a good place to look.
Well, a plot could involve Adam being sent on a mission by some other powerful entity in Hell to infiltrate the Hazbin Hotel and disrupt Charlie's plans. However, he ends up falling for Charlie and has an internal struggle between his loyalty to his original mission and his newfound feelings. Or perhaps Adam has a special power that Charlie needs to help with a big event at the hotel, like a demon - friendly concert, and they have to work together closely to make it a success.
One plot idea could be Lucifer seeing a different side of Adam that no one else has. Maybe Adam shows kindness towards a lower - class demon, which makes Lucifer question his own views. Another could be that they team up to solve a mystery in Hell that threatens both the sinners and the native demons. And a third idea is that Adam tries to introduce Lucifer to more 'human' concepts like forgiveness, which Lucifer initially resists but gradually comes to understand.
Perhaps Adam has a special power that could either help or hinder Charlie's plans for the hotel. They could go on an adventure together to find a way to use his power for good. Along the way, they face various challenges that force them to rely on each other. This shared experience brings them closer and they start to fall in love as they realize how much they mean to one another.
Charlie would be a great character to pair with Adam. Charlie is all about redemption and Adam could either be a foil to her or be influenced by her positive attitude. Their interactions could be really interesting as they have different outlooks on the whole idea of sin and redemption in the Hazbin Hotel world.