Is it possible to delete games you have saved in training mode?

Sure.

Unless you specifically saved your games to a different folder, you can find them in this folder:

C:\Program Files\Ubisoft\Chessmaster 10th Edition\Data\Users\<Your Name>

Just go to that folder, find your games (saved as *.PGN or *.CMG files), and delete the ones you want to get rid of.

I can find the games in their but I cannot figure out how to delete them. I have tried right clicking the game I want to delete and it just sends me to the help menu. When I highlight on the game I want to delete what do I have to do to get rid of it?

You can't delete them from within the game.

Shut down Chessmaster, and use Windows Explorer to browse to that folder. Then you can delete them...