I found mention of Robert's drowning in both Falls City newspapers.
From the Falls City News (Thursday June 30, 1881):
"The body of a boy was found floating in the river, near Iowa Point, last Friday, which was recognized as that of Robbie, son of Charles Hergesheimer, who was drowned at Rulo a short time ago.It was in such a state of decomposition that it was buried at the place, but when cool weather comes, will be taken up and buried in the Rulo cemetery."
From the Globe-Journal (Saturday June 25, 1881), repeating an item from Thursday's Daily Journal(June 23):
"The body of Robbie Hergesheimer was found yesterday at Iowa Point - It will be buried near where it was discovered, until the parents get settled in their new home."
The papers disagree on when the body was discovered, the Globe-Journal indicates June 22, while the News says June 24.Iowa Point is about 15 miles down river in Kansas.