Dear Gordon,
Thanks for your message. We have contact before, did you receive my belowing reply? Regarding your question, if you hold British passport, no need to apply for visa when you visiting Hongkong, but if you arrive mainland China firstly, then visit Hongkong, fly back to Beijing after the visiting to Hongkong, you need to apply for double-entry visa.
How do you think of our proposal? Is there anywhere I need to amend it for you? Please advise.
Hope the above is clear and helpful.
Should you have any additional questions, please feel free to contact us.