When checking the fuel pressure it could be low but doesn't always mean it's the pump. That's just one step of the trouble shooting. As you are doing, be sure it's getting enough voltage back there. If low it could be any number of things from corrosion to a bad switch relay etc. It seems you would also have lower voltage output at idle from the alternator which could make things even worse. On an older vehicle you could have corrosion growing far back up a wire under the insulation.
Just like an o2 for example. It doesn't mean it's absolutely the o2 sensor, it can be anything in that circuit. I have fixed a couple heated o2 sensor codes by simply removing and reinserting the fuse in the power center. The ecm picked up the resistance of the dirty oxidized fuse blades.