The History of the Putnam Family seems to be a definitive source. Look at these:
https://archive.org/details/ahistoryputnamf01putngoog/page/n112 (Nathaniel 1619)
and
https://archive.org/details/ahistoryputnamf01putngoog/page/n118 (John 1627 m. Rebecca)
These two are brothers, sons of John Putnam Sr and Priscilla Gould.
Both had a son John, and that's where the error lies on the Geni tree.
The John who is the husband of Hannah Cutler is the son of Nathaniel, not the son of John. I added Hannah Unknown, whom the Putnam Family History has as the wife of this John. Hannah Cutler needs to be moved to the other John (son of Nathaniel).
If you follow the genealogy in the book, you can line up the children with the correct parents.
There are a few that need to be added.
To Nathaniel, add (move) John who married Hannah Cutler, and add Joseph.
To John and Rebecca, add John 7/14/1667 who m. Hannah unknown, and Elizabeth b. 1665.
There may be other errors in the tree, but these are the ones I found. The profiles in question are locked, so I can't correct them. Could you please make the changes?