Learn Arabic (Modern Standard): تكلم المدير مع الموظف سرا. - The manager spoke secretly with the employee.

The noun for secret is سر. To turn it into an adverb of manner meaning secretly, we put it in the indefinite accusative case. For most words, this means adding an alif to the end, which changes سر into سرا (pronounced sirran).

Does the adverb سرا always have to go at the very end of the sentence?

Usually, yes. In Arabic, adverbs of manner in the accusative case like سرا naturally fall at the end of the clause, coming after the verb, the subject, and any prepositional phrases like مع الموظف.

Why does the verb تكلم start with a taa if it is in the past tense? Doesn't a taa prefix mean present tense for she or you?

That is a very common mix-up! While a taa prefix does mark the present tense for she or you, here it is actually built right into the past tense verb pattern. The verb تكلم is a Form V verb, and those always start with a taa in the past tense. Because the subject المدير is masculine, we know this is definitely a past tense verb meaning he spoke.

Does using مع change the meaning compared to using إلى after the verb تكلم?

Yes, it changes the nuance slightly. Using مع (with) shows that it was an interactive, two-way conversation between the manager and the employee. If the sentence used إلى (to), it would imply more of a one-way speech where only the manager was talking.