Buying a better graphics card and a CPU cooler is the best option here. The FX 6300 can handle midrange cards just fine, we're talking 970 /stock 980 levels here, but with something like a 1070 or 980ti there will be a roughly 30-35% bottleneck in most applications.
Your best bet is to go for a mix of the two.
You can get a surprisingly good upgrade for the price with 500 pounds, although this does go over budget by 70, bear with me as it is well worth it given your situation. Short of buying a new mobo, CPU and ram, this is the most cost effective option. This CPU will last you until this entire build becomes irrelevant, as this is one of the most powerful CPUs that there is for that motherboard.
So until you manage to get the funds for your next big upgrade, this will do you well.
PCPartPicker part list:
http://uk.pcpartpicker.com/list/Jg4LD8
Price breakdown by merchant: http://uk.pcpartpicker.com/list/Jg4LD8/by_merchant/
CPU: AMD FX-8350 4.0GHz 8-Core Processor (£135.99 @ Amazon UK)
CPU Cooler: be quiet! SHADOW ROCK LP 51.4 CFM CPU Cooler (£29.99 @ Amazon UK)
Video Card: Gigabyte GeForce GTX 1070 8GB G1 Gaming Video Card (£406.41 @ BT Shop)
Total: £572.39
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2016-07-03 15:29 BST+0100
DO NOTE however that you will want to get a 1080p 144hz monitor to get the most out of these parts, as it will most likely be wasted on 60hz 1080p, as will any upgrade up from what you currently have.
A good entry monitor is the BenQ XL2411Z or the Asus VG248QE, both of which give great quality at an affordable price.