How do you think that having this compact might have affected the Pilgrims once they arrived in North America?

By context, you refer to the Mayflower Compact.

This Compact remained in effect until 1691 and tried to establish a self-government in America, being the first document attempting to do so. The Pilgrims arriving at North America would have "legal" dominance and liberty to settle and bound themselves to this New World. Arriving on September 16, 1620 the Mayflower, a British ship, with 102 passengers was responsible for this document, hence its name.