The most common and most likely answer to this crossword clue is the 2 plus 6 letter word SO THERE!
I've seen this crossword clue in The Guardian
S
O
T
H
E
R
E
!
Buddy explains!
The straight part
We know that "that will teach you!" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case that will teach you! = so there!.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.