He might be shocked at first. In many fanfictions, Adrien would feel a mix of disbelief, anger, and sadness. He has always known his father as a strict but loving parent, and finding out he's Hawk Moth, a supervillain, would completely shatter his view of him.
There are multiple consequences. Firstly, his relationship with his father will be forever changed. He'll no longer be able to trust him in the same way. Secondly, his relationship with Ladybug might be affected. She may be more cautious around him, not knowing if he'll side with his father. Thirdly, Adrien will have to face the public's reaction if the truth ever gets out. He could become a target or be shunned. All of these aspects make for a very intense and emotional storyline in fanfiction.
One idea could be that Adrien initially feels shocked but also a sense of relief. He might have always had a feeling that Marinette was special, and now it all makes sense. So he starts to view her in a whole new light, with more respect and admiration for her dual identity.
It could be Gabriel Agreste. He has been a suspect in the original series for various reasons. He is often absent and has a lot of secrets. So in the fanfiction, it would make sense if he was finally revealed as Hawk Moth.
Adrien could meet other mysterious figures in the lair. It could be an old acquaintance of his father's who has been helping with his secret plans. Or perhaps he'll come across a captive who was being held by his father for some unknown reason.
One idea could be that his dad initially reacts with shock and disbelief. He might start to question how he didn't notice all the signs before. Then he could try to use Adrien's identity as Chat Noir to further his own business or social standing in some way, creating a lot of internal conflict for Adrien.
Well, he might be really surprised at first. Then he could start to see Marinette in a whole new light. Maybe he'll become more interested in her as a person, not just as a friend.