One readily admits graduates' complete failure — making university embarrassed, initially (6-3)

I believe the answer is:

master-key

'one readily admits' is the definition.
(key admitting the user to multiple places)

'graduates' complete failure making university embarrassed initially' is the wordplay.
I cannot really see how this works, but
'graduates'' could be 'mas' (people with Master of Arts degrees) and 'mas' is present in the answer.
This may be the basis of the clue (or it may be nonsense).

I've seen this clue in The Guardian.

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