This has been puzzling me for some time... Why (once the turbines are turning and power generation has commenced) can't an electricity power plant power itself from some of the energy it produces? I'm sure the answer is in rudimentary physics (which is clearly not my strong suit!).

Sure you can use the power from it but you also have to remember that they aren't always turning.

What you're suggesting is a perpetual motion machine, which is impossible. There's a fundamental concept in thermodynamics that says no device can power itself, because no device is 100% efficient. To generate X watts of electrical power the generator requires input of greater than X watts of energy from an external source such as coal, gas, wind, hydro, or atomic energy. It can't simply feed itself, due to losses from things such as friction in the generator bearings, resistance in the windings, and inefficiencies in how electric current is generated.

The output will be always less than input,therefore efficiency is always less than 100%.