He'd run it with kindness and integrity. He would hire people based on their skills and not discriminate. For example, he might hire a werewolf as a chef if they're good at cooking.
Another possibility is a corrupted member of the Ministry of Magic. They might be trying to use Harry as a scapegoat to gain more power within the Ministry. They could be working with the Death Eaters or acting on their own, but they would be a significant obstacle for Harry as he tries to stay alive and free while on the run.
A plotline could involve Harry being on the run due to a false accusation, and during this time, he realizes he's pregnant. He then has to prove his innocence while also dealing with the physical and emotional challenges of pregnancy. There could be a sub - plot where the magical world is in chaos because of his disappearance and the mystery of his pregnancy.
In such a fanfiction, Harry Potter might be seriously injured or even die. This would completely change the course of the wizarding world. His friends would be devastated. Hermione might try to use all her knowledge to find a way to reverse the situation if possible, while Ron would be furious and determined to find out who was responsible.
Maybe he wants to escape the constant danger in the UK wizarding world. Voldemort's followers might still be a threat even after his defeat, and France could seem like a safe haven.
One possible reason could be that he feels unsafe at Hogwarts. Maybe there are threats from dark wizards within the school that the teachers are unable to fully protect him from. Sirius, being his godfather, represents a safe haven.
One idea could be that Harry discovers a hidden magical society that has been living in secret all along and they offer him shelter while he's on the run. Another could be that he teams up with some unexpected magical creatures who help him evade the Death Eaters. Maybe he finds a way to use time - turning magic in a new and unorthodox way to stay ahead of his pursuers.
I'm not sure about specific 'Pregnant Harry Potter on the Run' fanfictions. You could try searching on popular fanfiction websites like Fanfiction.net or Archive of Our Own. They have a vast collection of Harry Potter fanfictions, and by using the relevant keywords in their search bars, you might be able to find what you're looking for.
Perhaps in the fanfiction, Harry Potter runs away from Britain because he is disillusioned with the way the Ministry of Magic operates. There could have been a major event where the Ministry fails him or betrays his trust. Also, he may be seeking a place where he can explore his magic without the constraints and expectations that come with being 'The Boy Who Lived' in Britain.