There is a small but awesome local climbing community in Cusco. Pop in to the gym ( 7aescueladescalada.com/) or maybe even send them a message in advance of your arrival. The proprietors of the gym are the ones doing all the development and can give you beta/introduce you to partners.
It might be worth paying them to show you around for a day or two and then ask them to introduce you to some climbers.
If you get time I highly recommend checking out Pitumarca, which is a newish crag 2.5-3 hours outside Cusco. It's an amazing crag with high quality limestone and fun routes. It's at 14,000+ feet, so make sure you're acclimated before heading over there.
The 7a Escuela folks are developing Pitumarca and can help get you there as well. They've got a PDF showing the routes on their website.
Lack of Spanish will make the whole thing a bit harder, but all the local climbers know at least a bit of english and the community is super welcoming.
Also, if you want to contribute to the local community bring them some bolt hangers. They're hard to come by and these folks are putting up tons of routes out of their pockets.