The most common and most likely answer to this crossword clue is the 3 plus 6 plus 2 plus 1 plus 5 plus 4 letter word TWO SHAKES OF A LAMBS TAIL
I've seen this crossword clue in The Times
T
W
O
S
H
A
K
E
S
O
F
A
L
A
M
B
S
T
A
I
L
Buddy explains!
The straight part
We know that "A moment of extreme agitation in Leicester?" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case a moment of extreme agitation in leicester? = two shakes of a lambs tail.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.