das Eingeständnis
admission
The act of admitting a fact, a mistake, or guilt.
Nach langem Zögern kam von ihm das Eingeständnis, dass er gelogen hatte.
After long hesitation, he made the admission that he had lied.

admission
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admission
The act of admitting a fact, a mistake, or guilt.
Nach langem Zögern kam von ihm das Eingeständnis, dass er gelogen hatte.
After long hesitation, he made the admission that he had lied.

The noun 'Eingeständnis' (admission) is derived directly from the verb 'eingestehen' (to admit). 'Eingestehen' means to admit a mistake or guilt.
Er musste schließlich eingestehen, dass er sich geirrt hatte.
He eventually had to admit that he had been mistaken.
What is being admitted can be expressed by a genitive or by a subordinate clause with 'dass' (that). The genitive construction is more formal.
Sein Eingeständnis des Fehlers kam für alle überraschend.
His admission of the mistake came as a surprise to everyone.
'Eingeständnis' (admission) is frequently used with the verbs 'machen' (to make) or 'ablegen' (to make, very formal). Typical adjectives are 'offen' (open), 'ehrlich' (honest), or 'schmerzlich' (painful).
Die Politikerin machte in der Pressekonferenz ein offenes Eingeständnis.
The politician made an open admission at the press conference.
recognition
'die Anerkennung' and 'das Eingeständnis' can both have to do with accepting something as true or valid. The important difference is in tone and context. 'Anerkennung' often sounds neutral or official, for example with a degree, a status, or a regulation. 'Eingeständnis' is used when someone admits a mistake, guilt, or an unpleasant fact. Learners may confuse the words because both express a kind of confirmation. In practice: for official rights or validity, 'Anerkennung' fits; for an admitted mistake, 'Eingeständnis' fits.

Sein Eingeständnis kam spät, aber danach konnte das Team den Fehler endlich korrigieren.
His admission came late, but afterward the team could finally correct the mistake.
Für die Anerkennung ihres ausländischen Abschlusses muss Sofia mehrere Dokumente einreichen.
For the recognition of her foreign degree, Sofia has to submit several documents.