It took me a couple of hours of (yes hours) playing this level on hardcore before I finally figured it out.
(I am giving this solution from what may be the first checkpoint? all I know is that I kept loading in front of the ark, while it was slowly moving around the right front of the enemy ship)
The initial thing you think you want to do is man the AA guns in the ark. This is incorrect.
Fire up the jet pack and notice that there are two arms coming off the enemy ship. On either side of each arm there are 3 rectangular pieces that come out of the arms with circular "on-board" mines in them. This are a total of 12 mines you need to destroy. Once these are destroyed, you have to "re-wire" and then destroy the power cores.
1) Line yourself up with the outside of the arm closest to you and hold the brake button (x) and fire away at the mines (the on-board mines, you dont have to destry the free-floating ones). You should be able to get all three mines in one or two passes.
**make sure you are lined up level with the arms, so that no other enemies/mines can fire at you
2) Line yourself up with the outside of the other arm (furthest from you when you load) and again, make a couple passes at the mines until you have destroyed all three
3)For the inside mines, line up close to the rear of the ship to the left or right of where the "cockpit" is located. From here you are only in the line of sight of one arms mines at a time. Destroy the side you have lined up on, and then switch sides and repeat.
**free-floating mines are released near this part of the ship, so make sure you listen for the proximity warning and if one gets too close, fly away from the ship, blow up the floaters and then return to the task at hand
4)After destroying all the "on-board" mines, fly to the front of one of the arms and press (B) at the two wire-boxes. The control rod will pop and and shoot it. Repeat on the other side and you are done.
**There are 3 or 4 AA guns on the enemy ship, but they never seemed to fire on me, just at friendly fighters...regardless they are easy to defeat between killing the "on-board" mines if you have to
**If you try to complete this mission (on hardcore at least) from the comfort of the AA gun turrets, the laser on the front of the enemy ship will kill you after you circle in front of it
**I would recommend killing all the on-board mines and then doing both power-rods, instead of one at a time, because it might throw off your rythym