CPU Fan - Help needed

Could I use a splitter, so that the cpu fan gets power from the PSU, and there's also a 3pin which plugs into the motherboard, so the motherboard knows there's RPM?

As far as I'm aware, you should be having the opposite problem if anything, so long as the 3 pins that are connected are the right ones (it might be possible to connect the wrong 3 pins of the fan header). It's pretty hard to do (if it's possible at all) because the clip on the cable should match up with the slot section on the header.

The middle pin on a 3pin connector is the one that checks for the fan speed, the 4th (unused in this case) pin is to control the fan speed without changing the input voltage.

If the fan is spinning, then it's likely you have connected it correctly, which means that either it's controlling it's speed via voltage (as it is designed to) or it's stuck at 100% (which might be annoying, but won't be causing your issue).

Before worrying about anything else, I'd double check how well the CPU is seated. If the contact isn't perfect it might explain your issue, and it's definitely worth checking out.
Oh, it'd also be good to know the model of your motherboard (if you can find it). If it has another 3pin fan header you could try the cooler in that to see what happens. Unless you fiddle with the BIOS, that fan should run at 100% speed.

My motherboard is a M2N68-LA (Narra6) according to HP's specifications website. I can't find any other fan sockets apart from the chassis fan one.

The fan is not currently running at 100%, as in speedfan, it fluctuates between 840-1240 RPM (I wish I could just keep it at a higher RPM, as it's not at all loud at the moment).

As far as I can tell, the heatsink is central over the CPU, and thermal paste shouldn't be an issue as it came pre-applied on the heatsink.

I've got a spare CPU fan which I've installed to see if the previous one was faulty, but there's basically the same results; idle around 53C.

Thanks.

Yeah, try that fan header. Googled images seems to show one pretty near the centre of the board. If the fan is plugged into that it should just spin at 100% speed. If that produces an identical temperature, then it's likely that something else is going on.

53degrees at idle is definitely hotter than you'd expect, but it's not especially high. That's the kind of temperatures you'd expect if the processor was working reasonably hard at something, so you should check how busy it is, in case some processes you aren't expecting are working it hard.

One thing that would definitely help you understand the problem would be to get a better idea of the temperature delta. If you start high but barely change under load, then you are probably fine. If your computer starts crashing when you push it, or starts hitting 70+, then it definitely suggests something is seriously wrong.

I tried plugging the fan into the Chassis Fan socket, but it still spins at 850ish RPM. I've got the fan from my old heatsink/fan which is 3 pin which I could possibly try (if only I could unscrew these insanely tight screws).

Speedfan showed it at around 25% load (just after booting up), and the temperature rises to around 50-55 degrees.

I'll try putting it under load later.