In the UK a power of entry is a statutory right for a person (usually a state official such as a police officer, local authority trading standards officer or a member of enforcement staff of a regulatory body) to legally enter defined premises.
What kind of things grant anyone such a right?
(Public officials in the UK have the power to demand entry to your home without a warrant to make sure illegal hypnotism isn't taking place.
They can also demand to inspect your pot plants, under a plant health order.
They can demand to see your refrigerator, to make sure it's at the correct setting.
And they can go into your garden, to check the height of your hedge.