That is very hard to say, as diagnostically speaking the random power off and only "Kernel Power" entry in the event log can be caused by such a huge list of things, and that includes both hardware or software related issues. My first thought would also be power supply. Standard ATX power supplies can be tested with relatively good accuracy using a cheap PSU tester to see if they are experiencing the majority of the failures experienced by a PSU. However, a server PSU can't just be connected into these as easily. About the only option here would be purchase another PSU and try that.
However, that may very well not be the issue just like you are stating. It could be motherboard issues as well, which is much more difficult to determine. But I know of software issues that can cause this, even firmware or driver versions needing to be updated. I know that if you contact into HP on it, that's going to be one of the main things they ask for is a report from the iLO system status or health status reports, and if you have all the latest firmware for your hardware devices.
I do believe the ML350 G6 still has iLO capabilities, so have you tried to log in there and see if there is any kind of warnings or hardware related errors listed in there or the associated health reports? I have a newer ML310e G8 Version 2 server recently that was doing the same thing, but it would only power off randomly (sometimes twice a day, sometimes once a week.) I checked the hardware logs using iLO and while that did show some errors, nothing related or associated to any hardware actually causing it, only reacting to the failure (sudden power loss.) However, it did show me that there was a detected error, and that was my iLO firmware was behind the most current recommended. Sure enough, after contacting HP for confirmation, they have reported that certain older firmware versions of iLO had been causing this exact issue and suggested upgrading the firmware.