After the power goes out, you manage to find a way back to your ship. Adam tells you to find a path to the auxiliary power system.
Find the Auxiliary Power Station
After you leave your ship, go to the left until you see a Keyhunter. Use a Power Bomb in this room to reveal a path to the left. Go through.
Roll into a ball and go to the left to go past the empty cocoon. In the area down below, go to the right and shoot the right side of the floor, and fall down the hole there. Go through the door down below.
In the Reactor Core room, make your way to the upper right door of the room and go through it. (The Save Room in the bottom right corner is not operational.)
Charge up your beam and go to the right, and shoot the Space Pirate with two charged shots (or three missiles) to kill it. Then go to the right and kill another Space Pirate. Go to the right and take the Energy Tank, then go to the right to reach the next room.
Go to the far right and shoot the floor one square to the left of the right wall. Fall through the hole, then kill the Space Pirate to the left. Go to the far left and shoot the floor one square to the right of the left wall, then go toward the right and kill the Space Pirate there. Go to the far right and shoot the floor one square to the left of the right wall. Fall through the hole, then kill the Space Pirate to the left.
Then go to the left side of this area and crouch, then shoot the left wall to break two blocks. Roll through the tunnel and take the Missile Tank at the end of the path. Then roll back through the tunnel, and shoot the floor near the right side of the tunnel to open a hole. Go down and kill the Space Pirate. Then shoot the floor near the right wall and go down the hole. Then shoot the floor in the middle of this area and roll into the tunnel. Roll to the right to fall through a Pit Block. At the bottom, drop a bomb, then roll through the hole. Go to the left and climb up the ladder, then shoot to the right and roll through the tunnel on the right. Drop a bomb at the dark grey block near the middle of this tunnel, then go up through the hole and go to the left to kill the Living Door. If you need to avoid its membrane, go to the right to go down the slope, and crouch to avoid the membrane as it flies overhead.
After the Living Door is dead, get the red X that it drops, go through the door, and fall to the bottom of the area. Roll into a ball slightly to the right of the middle of the room. You will hear a crash, three flames will fall from above, one at the far left, one in the middle, and one at the far right, and Yakuza will appear from above.
Yakuza can be difficult if you are not patient.
If you positioned yourself correctly, you will be to the right of the center flame that Yakuza shot from above. Stay rolled into a ball. Yakuza will crawl diagonally along the wall from the top left corner. When the center flame gets lower, but before Yakuza comes too close, ball jump to the left and approach the left wall. If you do not move from the right of the center flame, Yakuza will grab you in its pincers and drain a lot of your energy. Since Yakuza drains so much energy from you, you want to avoid this at all costs.
In the first half of the battle, stay in the left side of the room, staying rolled up as a ball unless you are sure Yakuza will not be able to reach you. When Yakuza stops, sometimes it will open its mouth and shoot three flames, one to the left, one straight down, and one to the right. Stand up when you see Yakuza stop, and run underneath it. When it opens its mouth, rapidly shoot Missiles up into it, then before it spits out its flames, roll into a ball to one side of where the center flame will fall, whichever side you think will be safest to avoid getting touched by Yakuza. If you realize you have to jump over the center flame, take care not to get in Yakuza's path. Avoid letting Yakuza touch you at all costs.
Eventually, Yakuza's legs will fall off and it will start spinning rapidly and shooting debris down as it bounces around the room. Yakuza will open its mouth when it reaches the peak of its jump, but it will also spit out some debris. Try to shoot missiles upward into Yakuza's mouth while also shooting or avoiding the debris.
After you hit Yakuza in the mouth enough times, it turns into a Core-X. You can shoot the Core-X with your regular beam to make it release free-floating X if you need to replenish your energy and weapons. After you kill it, take the Core-X and you will recover your Space Jump ability.
To Space Jump, you have to do a Spin Jump (i.e. jump to the left or right instead of jumping straight up), then press the jump button again when you are falling at maximum speed. To Space Jump more quickly, don't hold the jump button for very long. This will allow you to reach maximum falling speed sooner, because longer jumps cause you to slow down more at the peak of your jump. Shorter jumps cause less slowdown at the peak of the jump.
Go to the top of the room. There is a hidden tunnel to the left of the exit hatch. The tunnel must be entered from below. Go to the left through this tunnel to reach the Auxiliary Power Station. Stand in front of the console to activate Auxiliary Power.
Go through the door to the left to enter a Navigation Room. Adam says that you can use Save and Recharge rooms again, but not hatches or elevators. You have to find the source of the vegetation that is preventing the Reactor Core from working.
Go through the door on the left, shoot the light to open the gate, and go through the door on the left to reach the Reactor Core.