Degree given to redcoat in error (9)
I believe the answer is:
doctorate
'degree' is the definition.
(doctorate is a kind of degree)
'to redcoat in error' is the wordplay.
'in error' indicates anagramming the letters.
'to'+'redcoat'='toredcoat'
'toredcoat' anagrammed gives 'DOCTORATE'.
'given' acts as a link.
(Other definitions for doctorate that I've seen before include "Highest degree in university" , "Higher degree" , "High degree" , "Academic qualification" , "PhD or D.Phil" .)