The reason that the Pilgrims came is because they wanted religious freedom. They didn't have religious freedom in England. They had to be a part of the England church, they were ruled to do this by King James. They couldn't believe in what they wanted to. That is why they made a settlement in North America.
The Pilgrims faced a lot of problems, one of many is that they had rough seas. They didn't only have this problem on their boats though, a lot of people got diseases from rats and had starvation. One disease that these people had is that they had Scurvy. Scurvy is a disease where you don't have enough Vitamin C. Once they got there though they still had problems. They had harsh winters, half of them got sick and died from hypothermia, and on top of that they had to compete with Native Americans for resources.
I would fix the amount of food. What I mean by this is that they didn't bring enough food. I would also make it so they had blankets to keep them warm during the winter, so they didn't get hypothermia. I would also try to make a way so they could have fruits and vegetables (Vitamin C) so they didn't get Scurvy.