You likely have to come back on the CPU because a motherboard that can run the 8320/8350 well doesn't tend to be of the budget variety.
PCPartPicker part list /
Price breakdown by merchant
CPU: AMD Athlon X4 860K 3.7GHz Quad-Core Processor ($76.98 @ Newegg)
CPU Cooler: Cooler Master Hyper 212 EVO 82.9 CFM Sleeve Bearing CPU Cooler ($24.89 @ OutletPC)
Motherboard: Gigabyte GA-F2A88XM-D3H Micro ATX FM2+ Motherboard ($55.98 @ Newegg)
Memory: G.Skill Ripjaws Series 8GB (2 x 4GB) DDR3-1600 Memory ($41.99 @ Newegg)
Storage: Seagate Barracuda 1TB 3.5" 7200RPM Internal Hard Drive ($46.89 @ OutletPC)
Video Card: XFX Radeon R9 280X 3GB Double Dissipation Video Card ($194.98 @ Newegg)
Case: Deepcool TESSERACT SW ATX Mid Tower Case ($43.98 @ Newegg)
Power Supply: XFX 550W 80+ Bronze Certified ATX Power Supply ($26.99 @ Newegg)
Total: $499.68
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2015-09-25 12:36 EDT-0400
I'm assuming you have or don't need an optical drive. If you need one and don't have one, it's like $12 on top.
With 8320/8350 out of the mix without going real cheap on the motherboard and downgrading the GPU, I switch over to an FM2 build since this is for gaming. The 860K will perform as well in gaming as an FX-6300 will.
The PSU I supplied is much better and can fit into the budget thanks to the discount - that's a standard recommendation at the $60 tier. That's a power supply that will be good through future upgrades. That sale is only until the 30th, so if you buy after that, you may have to swap in either an entry-level PSU (like the original EVGA one you posted or a Corsair CX) or come down on the GPU to make room in the budget.
I cut out the extra thermal paste. We're already at a very tight budget and the thermal paste that comes with the hyper 212 is just fine. Realistically, there's no much difference between legit thermal pastes. You could use mayonnaise and it would still transfer the heat from the CPU to the heatsink to an extent. NOTE: THIS HAS BEEN TESTED BUT DO NOT TRY THIS.
That leaves enough budget for a really good GPU. That's the most important component for gaming after having a safe power supply.
At launch, A88XM motherboards frequently needed BIOS updates for Kaveri CPUs, but that was more than a year ago and the ones with older BIOS versions *ought* to be mostly out of the mainstream food chain by now (Gigabyte released a specific 3.1 revision that supports Kaveri completely out of the box). In the slight chance that it isn't, you can either call Gigabyte and swap in one with the BIOS upgrade or take it to your local PC shop where they will have enough CPUs hanging around to flash it for you quickly.