Learn Italian: L'ente ha erogato la sovvenzione mediante il pagamento. - The institution disbursed the grant by means of the payment.

Mediante translates to "by means of" and belongs to a formal, administrative register. Since this sentence uses bureaucratic terms like ente (institution) and erogato (disbursed), mediante fits the tone perfectly. While con (with) is grammatically correct, it sounds too casual here.

Could we use attraverso or tramite in this sentence instead of mediante?

Yes, they are highly synonymous in this formal register. However, tramite often implies the use of an intermediary (like a specific person or agency), while attraverso carries a slight nuance of going "through" a longer process. Mediante strictly and formally highlights the direct method or instrument used, which in this case is the payment.

Does mediante ever combine with the definite article like other prepositions do?

No, mediante never merges with definite articles to form an articulated preposition (like di becoming del). You must always keep them separate, which is why it is written here as two separate words: mediante il.

I don't recognize the verb erogare. Is it common in spoken Italian?