The only information I have re: Robert Sprout comes from memoirs written by my parents. The first historical reference to Robert appeared in 1652 (he would have been about 20 years old) when he and another Scituate resident were chosen to care for the destiute widow of an original settler. Records indicate that Robert was the son of Sir James Sprout of Kelfield, England.
Robert married Elizabeth Samson, whose father was a Mayflower passenger, in 1660. They moved to Newasket, an indian village and trading center. He, and other white settlers, purchased land there and eventually gained control of the settlement and incorporated it as Middleboro. They built a fort (the indians were not too happy with their presence)and took refuge there during King Philip's War (1675-6). They escaped from the fort, fled to Plymouth, and returned later to rebuild Middleboro.
Robert served in the 1690 Canadian expedition and returned to Middleboro, where he died in 1712 (at age 80). He owned considerable land in Middleboro and a farm in Scituate.
Your reference to frequently visiting the Scituate library caught my attention because I have thought about visiting Scituate to learn more about Robert Sprout and his family. Apparently, that "sourse" has little to offer.