Yes, you can run a home care agency from home in England, if your home is genuinely where you run it. CQC’s location guidance says a location is not "a private home, where that private home is not the primary place from where the service is managed", so a home that is your main base can be your registered location.
What CQC will not accept is an address you do not work from: "We do not consider virtual offices to be a location." PO boxes are out too, and the office must be in England.
Whatever the premises, you send evidence of legal occupancy with the application: the title register, a signed tenancy agreement or a signed licence agreement, plus written permission from your landlord, mortgage provider or equity release provider if you do not fully own it. "If you do not provide this evidence, we will not be able to approve your application."