When you reach the top of the core (before the cutscene) stand off to the sides and pick off the mutants and visceroids in the cells with your pistol. Then head into the cutscene.

When you can move again, dash behind the red cage Sydney's in, and the BH's shouldn't shoot you once you're behind it. You'll take quite a bit of damage while getting there. From behind the cage, pick off the stealth guys by strafing around. If Petrova starts shooting, duck back behind the cage until she stops.

When the black hands are gone, step out from behind the cage and run around facing the ramp until she shows herself by shooting. Paste her with a laser rifle so she stops and flails around while burning, and charge her until you're up close. When you empty your current laser magazine, switch to the flamethrower and she should be easy work.