Does Nyxbloom Ancient replace the entire mana ability or just the part that produces mana?
Nyxbloom Ancient only modifies the mana produced. Empowered Autogenerator still gets a counter.

Rule 106.12b says


  A replacement effect that applies if a permanent “is tapped for mana” or tapped for mana of a specific type and/or amount modifies the mana production event while such an ability is resolving and producing mana or the specified type and/or amount of mana.


Empowered Autogenerator's activated ability has two separate instructions: first you add a counter, then you add mana. Only the second part is the mana production event, so only that part gets modified by Nyxbloom Ancient's ability.