Can the power supply cause the computer to be difficult to turn on?

Yes. If the power supply is of a low wattage and has too many pieces of hardware connected to it, it can. Try buying a power supply with a wattage over 300.

One thing to check before replacing the power supply is the voltage selector in the back. If you use 120 volt power, then setting the power supply to 240 volts will cause the PC to only get half of the power it needs.

External peripherals with their own power supply will not impact the power supply or current in the computer. Printers and monitors usually get their power from the wall socket, not the computer, and the same goes for external modems that plug into the wall. Keyboards and mice take negligible power.