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Stock Problems/IssuesDiscussions related to warranty related issues and repairs, or other problems with the OEM parts and software for MINI Clubman (R55), Cooper and Cooper S(R56), and Cabrio (R57).
With out knowing what the codes were, this is a total shot in the dark. When you say one cylinder is low on compression, what do all four read? How many miles are on the mini?
Misfires can be caused by the High Pressure Fuel Pump. I've have several posts on this topic and my experience with it as well as the check engine codes it set off. The HPFP is covered up to 10 year or 120,000 miles.
Sounds like you need to do a leak down test and see where your compression leak is. Try to get a bore camera into the cylinder and see if there is damage to the piston/valves.
Either direction this goes you'll be into it for a head rebuild or a piston.
With 0 compression in number 1 you could do a leak down test to further determine the reason as Nick at Detroit Tuned pointed out. Either way, your next step is to pull the head. If you're lucky, one of the valves is cracked and you can get the head rebuilt.