***** NOT A TEACHER *****
Hello, Huacunli:
I did some googling and am delighted to share my findings.
(1) Sometimes "go on" = "continue."
(a) This is from Alexandre Dumas's The Three Musketeers:
"I did not go on, because it pleased me to stop here."
"She called out cooly to the coachman, 'Go on -- home!' "
(2) I also think that some people think that "Go on home" is a bit stronger than simply "Go home."
(a) Here is a sentence from the novel Think and Grow Rich by Napoleon Hill:
"[H]e yelled at her, 'I told you to go on home! Now go, or I'll take a switch to you [beat her with something].' "
(3) I can imagine this:
Two cars collide. Dozens of people run to the scene to look. Two police officers arrive. Officer Smith says, "OK, folks. We need you to
go home. Let the police and medical personnel do their jobs."
But the crowd does not disperse. So the other officer might yell "Didn't you hear Officer Smith? Now go on home right now. Get out
of here!"