A personal failing, with mistake finally admitted, for high-fliers (10)
I believe the answer is:
aeroplanes
'fliers' is the definition.
(I've seen this before)
'a personal failing with mistake finally admitted' is the wordplay.
'failing' is an anagram indicator.
'with' is a charade indicator (letters next to each other).
'finally' says to take the final letters.
'admitted' indicates putting letters inside.
The last letter of 'mistake' is 'e'.
'personal' is an anagram of 'roplanes'.
'a'+'roplanes'='aroplanes'
'aroplanes' going around 'e' is 'AEROPLANES'.
'for high' acts as a link.
This may not be correct. Some or all of it may belong to another bit of the clue.
Can you help me to learn more?
(Other definitions for aeroplanes that I've seen before include "Aircraft" , "Fliers" .)