Encourage, as in The crowd was cheering on all the marathon runners. Originating in the 1400s simply as cheer, this usage was augmented by on in the early 1800s.
Words nearby cheer on
The American Heritage® Idioms Dictionary Copyright © 2002, 2001, 1995 by Houghton Mifflin Harcourt Publishing Company. Published by Houghton Mifflin Harcourt Publishing Company.