Embarrassing error follows Best's header for Marsh (3)
I believe the answer is:
bog
'marsh' is the definition.
(a bog is a marshy area)
'embarrassing error follows best's header' is the wordplay.
'embarrassing error' becomes 'og' (I am not sure about this - if you are sure you should give a lot more credence to this answer).
'follows' means one lot of letters go next to another.
'header' indicates taking the first letters (in a down clue, the letter at the top of the word).
The first letter of 'bests' is 'b'.
'og' put after 'b' is 'BOG'.
'for' acts as a link.
Can you help me to learn more?
(Other definitions for bog that I've seen before include "convenience" , "Marsh" , "Soft muddy area" , "Swamp" , "Morass" .)