Keep pressing. Even if all you do is pillage his countryside down to the dirt, make it hurt.
Does that size 10 city in the north of the incan land have Versailles? That could be a nice target.
No. That's HC's Forbidden Palace.
But this does bring up a good point, imo.
Corihuayrachina has two things worth taking from HC:
Horses and his Forbidden Palace.
It doesn't appear the culture of Vilcamba or Machu Picchu will be enough to flip the city should it be kept. Taking/razing this city will eliminate
HC's only source of Horses and destroy his Forbidden Palace.
This deals a blow both to his military and to his GNP.
EDIT: After looking a bit more at the game,
Cuzco is NOT the city to target!
Given HC's religion and the amount of culture in the surrounding cities, you'll only be able to keep it if you take everything in the area. Additionally, it's not as great of a city as it appears to be on the surface, imo. (
Cuzco doesn't have a shrine as far as I can tell from the log .. only Isabella has them with Islam & Buddhism.)
If Cuzco falls, Machu Picchu will likely become the new capital. Not only is MP larger (with better trade routes); it has more

in the surrounding tiles. Given HC's running Bureaucracy, Machu Picchu will very likely make up a decent portion of the loss when it becomes the capital.
The only thing capturing Cuzco will do for you at this point is increase your maintenance costs and put you in contention for a perma-rebelling city.
If you want to take a city worth taking, target Machu Picchu.