The most common and most likely answer to this crossword clue is the 8 letter word SOLECISM
I've seen this crossword clue in The Times
S
O
L
E
C
I
S
M
More clues leading to the result SOLECISM
- One and a hundred is a thousand - that's wrong
- Fish is covered in cream after gutting error
- Only Charlie is beginning to make an embarrassing mistake
- One is in classroom erasing all content — embarrassing mistake
- Charlie is put in grave briefly – that’s a mistake
- Exclusive college disheartened intellectuals over minute breach of etiquette
- Smile, cos’ it could be a grammatical error
- A breach of etiquette*
- Starts to console injured suitor after single maiden’s gaffe
- Lone Conservative oddly in some error
Buddy explains!

The straight part
We know that "embarrassing mistake" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case embarrassing mistake = solecism.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does SOLECISM mean?
SOLECISM noun- An erroneous or improper usage.
- (grammar) Error in the use of language.
- A faux pas or breach of etiquette; a transgression against the norms of expected behavior.