Well, perhaps Harry is fed up with the false or malicious stories in the Daily Prophet in the fanfic and decides to take action to destroy it. Maybe he uses magic to burn all the copies or disrupt the printing process.
Hermione would most likely support Harry. She's always been his loyal friend and would see the injustice of the Daily Prophet's actions. She could help him research laws in the wizarding world (if there are any regarding slander or privacy) and also use her intelligence to find ways to counter the newspaper's claims.
Maybe Harry sues the Daily Prophet for spreading false rumors about him, like making up stories that he's secretly working for Voldemort again. It could be that these false reports are ruining his reputation and causing problems in his relationships at Hogwarts and in the wizarding community.
He might be interested in the articles about new magical creatures. Since he has had many encounters with different and often dangerous creatures, news about newly discovered ones or updates on known ones would surely catch his attention.
Maybe he wants to keep up with the latest news in the wizarding world. There could be important announcements, like news about Voldemort's activities or updates on the Ministry of Magic.
In a fanfic scenario, the photo in the Daily Prophet might be misconstrued. Suppose it's a photo where Hermione and Ron are just having a friendly chat, but the Prophet spins it as a romantic moment. This could lead to all sorts of drama. Ron might be embarrassed and Hermione could be angry at the false portrayal. Their friends would then have to step in to clear things up.
It often portrays their relationship as very strong. They are shown to support each other through whatever the Daily Prophet throws at them, whether it's good press or bad.
One interesting plot could be that the Daily Prophet reports on Harry and Ginny's adventures in the wizarding world post - Voldemort. Maybe they are on a mission to discover new magical creatures and the Prophet is covering their every move.
Hermione Granger is also very popular. In fanfiction related to the Daily Prophet, she might be working on a research project for the newspaper, using her vast knowledge to dig deep into important wizarding issues. Her intelligence and determination would shine through in this context.