R56 Hpfp?
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Yes. Cold start. And it doesn't want to idle when taking off slowly first thing in the morning. Usually keeping sport turned on help some. But ATM I'm having the valve cover replaced and trying to fix an issue with the temp sensor not reading after having the housing replaced.
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MINI has extended the warranty on the HPFP on some of the N14 engines to 10 years for 120K. You would need to call you local MINI dealer to see if your VIN is covered by this. They won't just replace it without running the service plan, which means you have to leave it for a few days for them to test under certain conditions.