APU vs CPU: Which is better for gaming?

Here's the best advice I can give you about CPUs: Clock rate and core count don't really mean jack -- at least not between different architectures. For example, a 4-core Intel i5-4690k at stock 3.5 GHz will best an 8-core AMD FX-9590 at stock 4.7 GHz nearly any day of the week. But when talking about just AMD, their 6 core FX-6300 at stock 3.5 GHz will also best the 4 core A10-6800k at stock 4.1 GHz most of the time too. This is because their architecture, Piledriver, doesn't really sport physical cores. Rather, they sport 'modules', with each 'module' being made up of two rather incomplete AMD 'cores'. So basically, a "4 core" AMD CPU is, depending on the task, closer to a 2 core CPU, and a 6 core AMD CPU is, again depending on the task, closer to a 3 core CPU. That's one reason the FX-6300 is more powerful -- that and the fact that the FX-6300 has L3 cache whereas the A10-6800k doesn't.

As for "how this [the A10] power is split up" well, it doesn't really work like that. The CPU and GPU don't "share" a clock rate, because they're not the same device. Unlike AMD's 'cores', the APU's GPU and CPU are individual components -- they just exist on the same die and operate mostly independently from one another aside from having to share resources from the same memory pool. The actual clock rate for the GPU on the A10-6800k (the Radeon HD 8670D) is 844 MHz in case you were curious.

As for the more important answer to your question, the FX-6300 with a dedicated GPU, as everyone else has been saying, will be more powerful. Actually, if possible you should aim for an i3 CPU such as the i3-4130 which, depending on where you look, is only $5 USD more expensive than the FX-6300. 2 hyper threaded Intel cores able to address 4 concurrent threads can be a lot more powerful than 3 AMD modules. The i3-4130 buts heads with the FX-6300, but overall it's a better option, both in terms of performance and because you won't necessarily have to change your motherboard if you upgrade to an i5.

UPDATE: Here's an idea: If spending ~$120 on a CPU alone is going to strap your GPU budget, then an i3-4130's integrated HD 4400, while far from powerful, will get you an operational and fully functional computer on its own until you can afford a good GPU. For a lower power, cheap choice that's quite powerful (close to PS4 level power), try for the ~$150 GTX 750 Ti.