What could cause my outlets to not have enough power?
I am going to guess that the outlet that the lamp is plugged into is the same one each time you have tried this. 

You may want to try plugging the lamp into one of the other four outlets. If the lamp does not come on then read on here. 

It is clear that the lamp outlet is wired in a daisy chain manner such that power feed comes first to the lamp outlet and then continues on to the other four outlets. The most probable cause of the problem is that there has been a failure at the lamp socket. When you plug in the lamp the tines on the plug cause the metal material in the socket to move around such that the downwind sockets receive power. Remove the plug and the metal moves and opens up the circuit.

This strange behavior almost always happens when the electrical wiring is attached to the outlet using the back stab holes on the outlet. The fix is to replace the outlet with a new one. Attach the electrical wires using the screws on the side of the outlet. There are two good types. One where the wires wrap around the side screw. The other type is where the wire enters from behind and then tightening the screw clamps the wire into place. Both these used with copper wire can provide years or reliable service. 

On the other hand the back stab type connections where the wire is poked into a hole to simply be retained by a spring clip are noted for failure and should not be used.