One possible fault could be that the character development wasn't as deep as it could have been. It felt a bit rushed at times, leaving you wanting more depth.
I'm not sure specifically what your 'my fault full story' is about. Maybe it's about a mistake you made? If you could give me some more details, it would be easier to answer.
One popular 'Your Fault' Harry Potter fanfic could be about Harry being blamed for a magical mishap that wasn't really his doing. For example, if there was a spell gone wrong in Hogwarts and everyone wrongly accused Harry because of his reputation. Another might focus on a situation where Harry is made to feel guilty for a decision Dumbledore made, like in a situation where a plan backfires and Harry is seen as the one at fault. There could also be fanfics where Harry is wrongly accused of stealing a powerful magical artifact, when in fact he was trying to protect it.
There are several important themes. One is the idea of finding beauty and meaning in a difficult life. Hazel and Augustus find joy in each other and in small things like books and shared experiences. Love is a central theme, not just romantic love but also the love between friends and family. The theme of mortality is also prominent. Their cancers constantly remind them of their limited time, and they have to come to terms with the idea of death while still trying to live fully.
Perhaps the main idea of the 'My Fault' book story is the exploration of the consequences that follow when someone makes a fault. It could show how one's actions can have a ripple effect on their own life and the lives of others, and the process of dealing with those consequences.
I'm not sure which specific 'My Fault' book story you are referring to. There could be many stories with that title. It might be a story about someone taking responsibility for their mistakes, perhaps in a personal growth context.