A comment, my Claro phone didn't work anywhere in Cusco, reception was awful. Telefonica Movistar worked far better.
There are four authorised Movistar agents in Cusco. You might still have problems registering it though, might need to find a Peruvian to register it with their details.
Code:
CAVEX Cusco
Esquina de Maruri con Pampa del Castillo 315 - Cercado - Cusco
L - V: 9:00 am a 6:00 pm y Sábados: 9:00 am a 1:00 pm
Agencia Manti - Cusco
Plazoleta Limacpampa Grande 509 Urb.Centro Histórico Cusco
L - V: 9:00 am a 7:00 pm y Sábados: 9:00 am a 1:00 pm
Agencia Digital Comunicaciones - Cusco
Plaza San Francisco 128 Cusco
L - V: 9:00am a 6:30pm (de corrido), Sábados: 9:00am a 1:00pm
Multicentro Cusco
Av. El Sol N° 382 Cusco
L - V: 9:00 am a 6:00 pm y Sábados: 9:00 am a 1:00 pm