After making a long trip back to your ship, Adam asks you to activate auxiliary power.
Find the Auxiliary Power Station
Once out of your ship, exit left and continue left. You'll see a Keyhunter (one of the creatures that had been in those cocoons). Kill it, then use a Power Bomb in this room to open the way left. Exit left.
You can simply pass by the empty cocoon here, just roll into a ball and get past the barrier. Continue down below, and shoot the floor to the right to destroy it. Fall below and head through the door.
The large Reactor Core room is filled with some kind of vegetation. You want to get to the top right door of the room and go through (the Save Room in the bottom right corner is not operational).
In the next area, up ahead below the ledge is a Space Pirate, so be prepared to jump its beams as you shoot it three times with Missiles (or twice with charged shots). Once you've beaten it, there is an Energy Tank straight ahead along the path. Exit right.
Go to the far right here, and jump up and shoot below you to break through the floor and fall through. Slowly head left, prepared to jump, because a Space Pirate ahead will shoot beams at you. Kill it, then shoot the floor below you at the far left. Get on the ledge and slowly head right, because another Space Pirate is along the way. Kill it, then shoot the floor at the far right and drop down to a Space Pirate in a small area. Kill this one.
At the left side of the enclosure, crouch down and shoot a beam to break open two holes in the left walls. Roll through as a ball to reach a Missile Tank at the end. Then return to the enclosure and shoot below you at the far right, and drop down.
Bomb the floor to the right of the little wall below you to open a hole next to it. Drop down, then roll right and you'll fall through a Pit Block. Lay a bomb at the bottom to open the way to the left, and roll out. Continue left, climb the ladder to the top, shoot towards the right to break open a hole in the wall. Spin jump off the ladder to grab the hole and pull up into the narrow tunnel. Bomb slightly right of the middle of the tunnel to break a block above you. Jump out, and approach the Living Door.
To avoid the membrane it may shoot out, walk down the ramp to the right and crouch down, and it will pass over you. Kill the eye, grab the red X, and go through the door. Fall to the bottom of the area and roll into a ball slightly to the right of the middle of the room. You'll 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.
A tough boss. Keep practicing, don't panic, don't be too hasty, and try not to get too frustrated.
If you positioned yourself right, you'll be to the right of the center flame that Yakuza shot from above. Remain as a ball. Yakuza will crawl diagonally along the wall from the top left corner. When the center flame gets lower, BEFORE Yakuza comes too close, ball jump to the left over the center flame 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 an enormous amount of your energy. If Yakuza grabs you, you pretty much have to restart the game and start the battle over (going back through the Reactor Silo), because it takes away so much energy that you're almost guaranteed to be defeated. You may be able to survive if Yakuza sucks you dry one or two times, but anything more than that and you're doomed.
In the first half of the battle, stick to the left side of the room, remaining as a ball unless you're sure Yakuza won't be able to reach you. When Yakuza stops, sometimes it will open its mouth and shoot three flames, one towards the left, one straight down, and one towards the right. Stand up when you see Yakuza stop and run underneath of it. When it opens its mouth, shoot a ton of Missiles up into it, then before it shoots out the 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.
If you're careful, and avoid letting Yakuza touch you, after shooting it enough times its legs will fall off and it will start spinning rapidly and shooting debris down as it bounces around the room. Your best bet is to stay put, and shoot Missiles straight up when Yakuza is overhead, destroying the debris before it hits you and trying to get a Missile into its mouth. After hitting it in the mouth enough times, it turns into a Core-X. Kill it as usual, grabbing any free-floating X it releases (and if you're in bad shape, you may want to shoot it with your beam a bunch of times to make it release a lot of X to heal you up). When you've beaten it, collect the Core-X to recover a lot of energy and recover your Space Jump ability.
The only way out is up, and you have to use your new Space Jump ability to get there. Jump at an angle left or right to Space Jump, then after you've passed the peak of your jump, press the jump button again to float higher (it helps me to press left or right when I try to bounce higher, but that's probably not necessary). Don't try to just tap the jump button repeatedly; it won't work. You'll need to get a sense of the timing of Space Jumping.
Anyway, once you get the hang of Space Jumping, get to the top of the room. There's a hidden tunnel to the left of the hatch leading back to the path that brought you here. The tunnel must be entered from below, and it continues towards the left. Exit left through the tunnel to reach the Auxiliary Power Station. Stand at the console to activate Auxiliary Power.
Exit left to enter a Navigation Room, and uplink for your next briefing. Save and Recharge Rooms are active, but hatches and elevators are not. Adam tells you that you must find the source of the vegetation choking the Reactor Core to reactivate the power. Exit left, shoot the light to open the gate, and exit left once more to reach the Reactor Core.