A fifth teenager, Connor Loftus, has signed a similar deal following his release from Bury at the end of last season.
Central defender Burke, right-back Johnson, midfielder Sinnott and striker Leonard will now work closely with the club's development coach Steve Eyre, whose job it is to bridge the gap between junior and senior football.
"Being a professional doesn't guarantee you a career in first-team football and that is the next development of their careers," Eyre said.
"As a club we need to create a pathway which these players can walk down and hopefully take the opportunities that are handed to them."
