Earth Impact Mini-Game:
In the Reliance level, enter the building through the vents. You will find a room with guards and Arcade machines. Kill the guard playing the arcade and give it go yourself. From this point onward, you can play the mini-game at any time through the Pause Menu.
X Space '92 Mini-Game:
In the Madness level, go to the storage room. This room is before the 3rd checkpoint and is the door which requires a keycode to open with a guard in front of the door and a camera above. Use your Telekinesis to grab the game from a shelf in the storage room. From this point onward, you can play the mini-game at any time through the Pause Menu.
Thanks to Revolution readers frannykorn and CELEBUS!