Cornejo has been awarded grants and residencies from the Sharpe Walentas Studio Program, Creative Capital Foundation, New York Foundation for the Arts, Sculpture Space, The Center for Book Arts, Lower Manhattan Cultural Council, the British Council, Arts Council of England, Vermont Studio Center, the Henry Moore Institute and the Ministry of Education of Japan. He has participated in group exhibitions like the V Bienal Barro de América in Venezuela, The S-Files at El Museo del Barrio in New York, Busan Biennial 2008, the XII Havana Biennial in 2015 and Vacio Museal at MAC Lima in 2016.

Solo exhibitions have been held at galleries: Peruvian North American Cultural Center in Lima, Nomura Foundation Bellini Art Gallery in Yokohama, Gallery Jun, Gallery Gyokuei and Gallery Kobo Chika in Tokyo, Gallery Light Contemporary in London, Gallery Artco and Gallery Lucia de la Puente in Lima. He is represented by Gallery Lucia de la Puente in Lima Peru and Art Front Gallery in Tokyo.