Flushed and embarrassed, admitting wrongdoing after head of school's demoted (6)
I believe the answer is:
rinsed
'flushed' is the definition.
The answer and definition can be both related to bodily activities as well as being past participle verbs.
Perhaps they are linked in a way I don't understand?
I can't explain the remainder of the clue.
(Other definitions for rinsed that I've seen before include "Got rid of soap" , "Swilled" , "Washed out with clean water" , "Washed out with fresh water" .)