If you intend to run the circuitry from a battery, no adjustment is necessary. Just wire the relay contacts into the active (hot) conductor of your appliance.
Your relay must be a type with a 6V DC coil, and the contacts must be rated to your mains voltage (110/120/230/240V - depending on where you live), and have a current carrying capacity of at least whatever your appliance draws.
If you want to do away with the battery, and supply the circuitry from the mains, use this modified schematic:
(NB, I have written 240V on the primary of the transformer, however, if you live in a 110V/120V area, obviously, you should use a 110V-6V transformer. The transformer need not be large, only 500mA approximately)