I have the following code, however it gives me an error:

new public Game Game
{
     get { return base.Game; }
}

Error 1 Cannot implicitly convert type 'Microsoft.Xna.Framework.Game' to 'MyGame.Game'. An explicit conversion exists (are you missing a cast?) K:\MyGame\MyGame\ScreenManager\ScreenManager.cs 47 26 MyGame

Why not try and return "this" ?

Sorry, I misunderstood your question.

Error 1 Cannot implicitly convert type 'Microsoft.Xna.Framework.Game' to 'MyGame.Game'. An explicit conversion exists (are you missing a cast?

Your problem is that your main class is called Game. The default name, when created by the IDE is Game1, you probably changed that to Game, so now there is a conflict between classes names.

I've tried using the following code in a component in a new project, where the main class is named Game1, and it works

new public Game Game
        {
            get { return base.Game; }
        }

This might be also something to keep in mind, not sure if it applies to you.

// Creates class MyGame that inherits from the base class Game
class MyGame : Game   
{
   // returnsthis instance of your extended MyGame class   
   public MyGame  GameInstance
   {
      get { return this; }   
   }
   
   // returns the instance of the base class without your
   //    custom things you added to MyGame   
   public Game  BaseInstance
   {      
      get { return base; }
   }
}