Reserve team top scorer Lee - on target against Mauritius last week - has signed a one-year contract while midfielder Kieran, who enjoyed a 60-minute stint in that friendly, has two years.
'Barney', who scored 17 goals in 17 games for the reserves, said: "I’m going to work throughout the summer, try to get physically stronger and come back fighting fit and raring to go.
"Pre-season is very important. Last season I scored in the first friendly, felt good and sharp in training and then broke my collar bone in the last game. Hopefully I’ll get another chance and that’s when I can prove myself. I can’t wait for next season."
Lee Barnard and Kieran McKenna have both agreed new deals with the club.