Power, Voltage, Current Supplies
A load requires current to operate, and current requires voltage to propagate. Supplying power is simply shorthand for saying that current is being supplied and voltage is being applied to a given load.

So, saying "supplying power to x" is somewhat more accurate than saying "supplying current to x" because you can't really have current without voltage. (You can have voltage without current though, or infinitesimally small currents, when the voltage is insufficient to overcome resistance. But since current is required to do any work, it's rather pointless.)