The only real way to test any power supply is to use a ohm/voltmeter, and check each wire's voltage output. Since You want to use an Atx or derivative from that form factor you will have to use the adapter cable setup suggested and then determine which wires from the normal Atx pinout (green wire usually) are funneled through the adapter cable to the white connector. on the MB. Once you know which wire on your original PS is powered on then you can run to ground and test output voltages. The usual problem for these cheap junk power supplies That are used by ALL OEMs to save and make a profit when only the fans turn on, and there is no bootup process, is the power-good circuit is gone bad. (All quality PS's for boards must deliver power to the CPU within a time interval or the CPU will not BOOT!)
The
ATX specification defines the Power-Good signal as a +5-volt (V) signal generated in the power supply when it has passed its internal self-tests and the outputs have stabilized. This normally takes between 0.1 and 0.5 seconds after the power supply is switched on. The signal is then sent to the
motherboard, where it is received by the processor timer chip that controls the reset line to the processor.
The ATX specification requires that the power-good signal ("PWR_OK") go high no sooner than 100 ms
after the power rails have stabilized, and remain high for 16 ms after loss of AC power, and fall (to less than 0.4 V) at least 1 ms
before the power rails fall out of specification (to 95% of their nominal value).
Cheaper and/or lower quality power supplies do not follow the ATX specification of a separate monitoring circuit; they instead wire the power good output to one of the 5 V lines. This means the processor will never reset given bad power unless the 5 V line drops low enough to turn off the trigger, which could be too low for proper operation.
Most likely this is what HP engineers/ and motherboard suppliers, sicn HP does make any of their boards have done this with your 6300. If the power-good circuit is not the problem then the board or a component on it is not performing or is deceased. Since your board has no speaker pinout to test failure beeps you only option would be a board tester like this:
Tester