Private girl admits old embarrassing error (3,4)
I believe the answer is:
own goal
'embarrassing error' is the definition.
(I've seen this before)
'private girl admits old' is the wordplay.
'private' becomes 'own' (as in one's own or private belongings).
'girl' becomes 'gal' (synonyms).
'admits' is an insertion indicator.
'old' becomes 'o' (common abbreviation eg in OE for Old English).
'gal' going around 'o' is 'goal'.
'own'+'goal'='OWN GOAL'
(Other definitions for own goal that I've seen before include "Self-made disadvantage" , "(Soccer) score which helps opposing team" , "'Action that turns out bad for the person who took it (3,4)'" , "Disadvantageous move for the party making it" , "Self-inflicted wound" .)