The little old king’s sign of impatience (3)
I believe the answer is:
tut
'sign of impatience' is the definition.
'tut' can be an answer for 'impatience' (I have seen 'Express impatience ' mean 'tut' so perhaps 'impatience' could also mean 'tut'). I'm not certain of the 'sign of' bit.
'the little old king's' is the wordplay.
'the little' means to remove the last letter (the word is not quite complete).
'old king's' becomes 'tuts' (King Tut).
'tuts' with its final letter taken away is 'TUT'.
Can you help me to learn more?
(Other definitions for tut that I've seen before include "in short I don't like that!" , "King exhumed" , "Reaction" , "Sound disapproving" , "Show disapproval" .)