Former Exeter player Pope, 26, can also play on the wing and joined the Championship club in 2013.
May, 22, has made 69 appearances for the Penzance-based side, and has been their top point-scorer for the past two seasons.
Moyle, 24, can play at full-back, wing or fly-half, and made his Pirates debut in 2012.
"All three individuals have developed as players, and Laurence is now one of the standout 10s in the Championship," said coach Gavin Cattle.
Scrum-half Mike Pope, fly-half Laurence May and back Kyle Moyle have all signed new one-year deals at Cornish Pirates.