Game database- many to many relationship
Yes, I believe you are on the right track.  The gameDate and gameScore fields are more a part of the game entity and not the player, so they should be on the Gametable.  Besides, bridge tables should really only contain the related ID fields.

If gameDate and gameScore were on the PlayerPlaysGame table, they would only be dependent on the gameID and not the playerID.  By definition, this would constitute a partial dependency, and would violate second normal form.