Joseph Smith was born and lived much of his young life in the New England states. Calvinism had a strong influence in his day. Puritanism in practice was not strong during his life but Presbyterianism and other Calvinist religions were. Joseph's mother and a few siblings had joined with the Presbyterians before Joseph's First Vision. However, some of these Puritan doctrines had strayed from the truth; in fact all Christian religions had strayed to varying degrees from the truth - they all had much of the truth but the fullness was lost. Further, the organization of Christ's church was no longer on the earth. Because of this, it was necessary for a restoration to occur. The Restoration was not a reaction to Puritanism or any other contemporary religion. That's what the Reformation was - a reaction to the prevailing church and dogmas. The Restoration was something new; it was putting new wine in new bottles, or rather, old wine made anew and poured into new bottles.
Another way the Puritans helped pave the way for the Restoration was with their strong emphasis on personal Bible study, which meant that literacy was important. They had higher literacy rates than other communities within the New World. The Latter-day Saints had and have a similar focus on education; we believe education and life-long learning are important. The Puritans believed education was important for both men and women; within Mormonism, we have always placed a similar emphasis on education for all, even if it is only informal. The LDS Church today has a donation-funded Perpetual Emigration Fund that helps members around the world break the cycle of poverty by providing them with education loans that they can use to receive college or technical training. Puritanism and Mormonism are similar in other ways. The Puritans also emphasized fasting and prayer, just as we do in the LDS church. Going without food and drink helps us overcome the natural man as we pray and seek strength and guidance from our Father.
The Puritans placed a large emphasis on time. They often turned their whole houses into sundials because clocks were prohibitively expensive for most people; they could use the relationship between the sun and their houses as a means to tell time. The Puritans abhorred wasting time or being idle. People were even fined for "misspending their time" (p.158). They believed that it was best to arise early and retire early so that they waste as little time as possible. They even tried to decrease the amount of sleep they received in order to accomplish more, especially scripture studying or acting in the service of God (p.161). Does that sound like LDS teachings? "Cease to be idle; cease to be unclean; cease to find fault one with another; cease to sleep longer than is needful; retire to thy bed early, that ye may not be weary; arise early, that your bodies and your minds may be invigorated" (D&C 88:124). Additionally, throughout the Book of Mormon those who were wicked were often described with the following words: "And it came to pass that I beheld, after they had dwindled in unbelief they became a dark, and loathsome, and a filthy people, full of idleness and all manner of abominations" (1 Ne. 12:23). [As an aside, I want to point out that becoming a "dark" people is probably best interpreted as a reference to a darkness of countenance (i.e., not having the light of Christ shining in one's countenance or eyes; being full of darkness instead of light)]. Within both Puritanism and Mormonism there is counsel against idleness.
Puritanism likely had some degree of influence on the culture of Mormonism (culture is not the same as doctrine), just as it had (and has) on American culture. While some doctrines of Puritanism were similar to those of the LDS faith, there were significant differences, predestination was one of the biggest differences. However, differences aside, Puritanism helped pave the way for the Restoration.
Fischer, D. H. (1989). Albion's Seed: Four British Folkways in America. Oxford University Press, Oxford.