If you pay for care privately, and are not happy with the service you are receiving, you do not need to put up with it.
If there is something specific you are not happy with, the first thing to do is to speak informally with the manager. Hopefully, they will be able to address your concerns quickly and effectively. If they don’t, a written complaint would be the next step.
If you are not happy with the outcome after a written complaint, as a paying customer you have the right to choose where and how you spend your money. When getting in touch with other agencies, it is important to explain the reasoning behind the change, so you can be sure that they will be an improvement on the existing one.
Other external agencies to contact would is the local council, the Ombudsmen, and the CQC. Although the CQC don’t investigate individual cases they do bear complaints in mind when inspecting providers.