zugeben
to admit
To acknowledge or confess a mistake, guilt, or fact, often reluctantly.
Schließlich musste er zugeben, dass er sich geirrt hatte.
In the end, he had to admit that he had been mistaken.

to admit / to add
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to admit
To acknowledge or confess a mistake, guilt, or fact, often reluctantly.
Schließlich musste er zugeben, dass er sich geirrt hatte.
In the end, he had to admit that he had been mistaken.

'Zugeben' (to admit/to add) is a separable verb. In a main clause, the prefix 'zu' goes at the end. In subordinate clauses and in the infinitive with 'zu', the verb stays together.
Er gibt seinen Fehler offen zu.
He openly admits his mistake.
When you admit what you did or what is the case, a subordinate clause with 'dass' often follows. The conjugated verb of the subordinate clause then comes at the end.
Sie gab zu, dass sie das Dokument verloren hatte.
She admitted that she had lost the document.
In the meaning of 'hinzufügen' (to add), 'zugeben' (to add) is used almost exclusively in instructions such as recipes or chemical processes. So the context is cooking, baking, or mixing.
Geben Sie zum Schluss noch eine Prise Salz zu.
At the end, add a pinch of salt.
to recognize / acknowledge
'anerkennen' and 'zugeben' can both mean that you accept something as true, often although it is unpleasant. 'zugeben' is used especially for a mistake, guilt, or a fact that you perhaps did not want to say before. 'anerkennen' sounds a bit more matter-of-fact and can also mean that you accept a difficult truth. Learners may confuse the words because both have to do with admission. For personal guilt or a mistake, 'zugeben' is usually more natural. For a fact or a problem that you accept, 'anerkennen' often fits.

Sie gibt zu, dass sie den Termin vergessen hat.
She admits that she forgot the appointment.
Nach dem Gespräch musste er anerkennen, dass der Plan zu teuer war.
After the conversation, he had to acknowledge that the plan was too expensive.