Learn Arabic (Modern Standard): تعرق الرياضي في الماراثون تعرقا. - The athlete really sweated in the marathon.

This is the Cognate Accusative (المفعول المطلق). It is the verbal noun (masdar) of the sentence's main verb, تعرق (he sweated). By repeating the verb's action as an indefinite noun without any modifying adjectives, it strongly emphasizes that the action actually happened or was intense. In English, we often express this emphasis with words like 'really' or 'certainly.'

Are the verb تعرق and the noun تعرقا pronounced with the exact same internal vowels?

No, the internal vowels differ slightly because both are Form V words. The past tense verb is تَعَرَّقَ (ta'arraqa) with a fatha on the doubled 'raa'. To make the verbal noun, you change that fatha to a damma, making it تَعَرُّق (ta'arruq). It is written with an alif at the end here (تعرقا) to carry the accusative tanwin (-an) required for the Cognate Accusative.

If we wanted to say 'The athlete sweated heavily', how would the use of تعرقا change?

You would add an adjective after it, such as تعرقا شديدا (heavy sweating). When you add an adjective, the Cognate Accusative stops being 'for emphasis' and instead describes the type or extent of the action. The lesson here focuses on the emphasis form, which requires the verbal noun to stand completely alone without an adjective.

Could we just use كثيرا (a lot) instead of the Cognate Accusative here?