Unable to determine the principal end of the '' relationship. Multiple added entities may have the same primary key.

Change the Id reference to object reference.

---

Example:
var player = db.PLAYERS.FirstOrDefault(p=>p.PlayerId = 13);
var newPlayer = new DRILL_PLAYERS();

newPlayer.PlayerId = player.PlayerId;           <== remove
newPlayer.PLAYER = player;                         <== use that 

db.DRILL_PLAYERS.AddObject(newPlayer);
db.SaveChanges();