IT depends on your CPU. The additional 4 pins were added to support the higher power draw of newer (at the time) Core 2 Quads. The 4 pins will happily provide power to most lower end CPUs, but if you have a Quad I would double check the TDP and make sure you can provide enough power with the 75W that the 4-pin can provide and that you don't need the 150W that the 8-pin provides.
Either way, I would upgrade your PSU. You can get a good, brand name, high efficiency PSU with the proper connectors for well under $100 and you will be all set in the event that you upgrade down the road.
Edit: The Rana has a 95W TDP, this means that your CPU will consume 95W on max load. Your 24 pin power connector provides 144W max on the 12V rail. 75 of these Watts are used by PCIe, which leaves ~69W for the CPU (though other components will use some of this so in reality it's probably closer to 60W). This means that with the additional 75W from the 4-pin you can ~135W available to the CPU under max load. This should be plenty assuming you are doing no overclocking and your motherboard and you don't have any other major 12v draws in your system (like an additional video card).
Seriously though, you should still get a new PSU. They can be reused in newer computers if you upgrade and there's no guarantee that you aren't voiding your motherboard warranty even though it should work.
I happened to have a HP G600 standing right next to me from a customer, so I took some pictures:
As visible in the top picture, the blue and purple cables going into the 6 pin plug are not as thick as the other 4, so those would be the ones carrying the power on signal, with the 4 black and yellow ones carrying power. This is clearly not compatible with a standard PCIe 6 pin GPU connector.
I would advise you to check the HP support site for the HP ProDesk 600 G1 for a list of compatible graphics cards, that will work with your current PSU.
For example, I found this AMD Radeon HD 8490 DP (1GB) PCIe x16 Graphics Card on the list, which is a somewhat decent card, which is guaranteed to work with your system.
As mentioned in the other answers, trying to get a standard PSU to fit the case/motherboard would be a major hassle. I would definitely not recommend this.
The lesson here is: don't use a Professional Desktop PC for Home/Entertainment/Gaming purposes, if you can avoid it, as they are typically hard to upgrade.
EDIT: After reading @Jasons comment below, I did some further research. The Q85 Chipset features one PCIe x16 expansion slot.
From the HP support web site:
Expansion slots
3 PCIe x1
1 PCIe x16
And from the Intel Q85 Chipset specifications:
Supported Processor PCI Express Port Revision 3
Supported Processor PCI Express Port Configurations 1x16
This is obviously different from what I wrote in my hasty comment below.
Best Answer
I guess the upper pins (the terms "upper", "lower", "left" and "right" refer to your original photo) are connected with yellow cables and are supposed to transmit +12V. The bottom pins are connected with black cables and are designed for GND. Refer to the user manual of your PSU to confirm this.
The left connector (8) is the one you connect to the PSU. The right connector (4+4) is the one you connect to the motherboard. If you need to connect an 8-pin socket to an 8-pin socket, it shouldn't matter though.
Note a beveled pin fits any socket type. By making the 4+4 part the way it is, they made it possible to connect the right connector:
In other words the right part of the right connector is universal all-beveled design, so it can be used as the right part or as a (standalone) left part.