NOT A TEACHER
I found a similar sentence in a great book (IMHO) written by a fantastic grammarian * (IMHO):
"She ran into the house crying."
Do you agree that the sentence is pretty similar to your "The man went away limping"?
If you do, here is how the expert explains it:
The word "crying" modifies the verb "ran" and is thus an adverbial element.
This expert says that "crying" has relations with both the subject "she" and the verb "ran."
I believe that you could look at it this way:
"She ran into the house [while she was] crying." As you can see, "while she was crying" definitely modifies the verb,
and is thus adverbial.
*George Oliver Curme, A Grammar of the English Language (1931), Vol. I (Parts of Speech), p. 43.
Interested in Language