Gaming build on £500 budget

Solution
Build can definitely be better. That CPU is two generations back.

Intel -
PCPartPicker part list / Price breakdown by merchant

CPU: Intel - Core i5-8400 2.8GHz 6-Core Processor (£159.54 @ Aria PC)
Motherboard: MSI - H310M PRO-VH Micro ATX LGA1151 Motherboard (£51.99 @ Amazon UK)
Memory: Team - Vulcan 8GB (2 x 4GB) DDR4-3000 Memory (£78.27 @ CCL Computers)
Storage: Seagate - BarraCuda 1TB 3.5" 7200RPM Internal Hard Drive (£29.99 @ Aria PC)
Video Card: MSI - GeForce GTX 1050 Ti 4GB Video Card (£138.35 @ Amazon UK)
Case: Rosewill - SCM-01 MicroATX Mini Tower Case (£21.34 @ Amazon UK)
Power Supply: Corsair - CXM...
Build can definitely be better. That CPU is two generations back.

Intel -
PCPartPicker part list / Price breakdown by merchant

CPU: Intel - Core i5-8400 2.8GHz 6-Core Processor (£159.54 @ Aria PC)
Motherboard: MSI - H310M PRO-VH Micro ATX LGA1151 Motherboard (£51.99 @ Amazon UK)
Memory: Team - Vulcan 8GB (2 x 4GB) DDR4-3000 Memory (£78.27 @ CCL Computers)
Storage: Seagate - BarraCuda 1TB 3.5" 7200RPM Internal Hard Drive (£29.99 @ Aria PC)
Video Card: MSI - GeForce GTX 1050 Ti 4GB Video Card (£138.35 @ Amazon UK)
Case: Rosewill - SCM-01 MicroATX Mini Tower Case (£21.34 @ Amazon UK)
Power Supply: Corsair - CXM (2015) 450W 80+ Bronze Certified Semi-Modular ATX Power Supply (£47.99 @ Amazon UK)
Total: £527.47
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2018-08-22 13:12 BST+0100

It is over budget, but it is much more powerful than your system and will serve you better.

Ryzen -
PCPartPicker part list / Price breakdown by merchant

CPU: AMD - Ryzen 3 2200G 3.5GHz Quad-Core Processor (£84.98 @ PC World Business)
Motherboard: ASRock - A320M-HDV Micro ATX AM4 Motherboard (£43.97 @ Amazon UK)
Memory: Team - Vulcan 8GB (2 x 4GB) DDR4-3000 Memory (£78.27 @ CCL Computers)
Storage: TCSunBow - X3 240GB 2.5" Solid State Drive (£36.99 @ Amazon UK)
Storage: Seagate - BarraCuda 1TB 3.5" 7200RPM Internal Hard Drive (£29.99 @ Aria PC)
Video Card: MSI - GeForce GTX 1050 Ti 4GB Video Card (£138.35 @ Amazon UK)
Case: Rosewill - SCM-01 MicroATX Mini Tower Case (£21.34 @ Amazon UK)
Power Supply: Corsair - CXM (2015) 450W 80+ Bronze Certified Semi-Modular ATX Power Supply (£47.99 @ Amazon UK)
Total: £481.88
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2018-08-22 13:14 BST+0100

Underbudget and has SSD and HDD. Just make sure to update the BIOS on the motherboard at a local computer shop or contact AMD for a boot kit before putting the new CPU.
 
Solution
I'd dump the ssd & go with the 1500x.

PCPartPicker part list / Price breakdown by merchant

CPU: AMD - Ryzen 5 1500X 3.5GHz Quad-Core Processor (£124.79 @ Aria PC)
Motherboard: MSI - B350M PRO-VDH Micro ATX AM4 Motherboard (£59.99 @ CCL Computers)
Memory: Team - Vulcan 8GB (2 x 4GB) DDR4-3000 Memory (£78.27 @ CCL Computers)
Storage: Seagate - BarraCuda 1TB 3.5" 7200RPM Internal Hard Drive (£29.99 @ Aria PC)
Video Card: MSI - GeForce GTX 1050 Ti 4GB Video Card (£138.35 @ Amazon UK)
Case: CiT - CIT-F3WHITEBLACK MicroATX Mini Tower Case (£25.99 @ AWD-IT)
Power Supply: Corsair - CXM (2015) 450W 80+ Bronze Certified Semi-Modular ATX Power Supply (£47.99 @ Amazon UK)
Total: £505.37
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2018-08-22 15:20 BST+0100
 
old gen I5 is not recommended, my take:

PCPartPicker part list / Price breakdown by merchant

CPU: AMD - Ryzen 5 2600 3.4GHz 6-Core Processor (£155.79 @ Alza)
Motherboard: Gigabyte - B450M DS3H Micro ATX AM4 Motherboard (£67.20 @ Aria PC)
Memory: Team - Vulcan 8GB (2 x 4GB) DDR4-3000 Memory (£78.27 @ CCL Computers)
Storage: Toshiba - 1TB 3.5" 7200RPM Internal Hard Drive (£34.00 @ AWD-IT)
Video Card: MSI - Radeon RX 560 - 1024 4GB AERO ITX Video Card (£110.68 @ Amazon UK)
Case: Thermaltake - Versa H22 Window ATX Mid Tower Case (£32.98 @ Ebuyer)
Other: Corsair CP-9020097-UK VS550 ATX/EPS Vs Series 80 Plus Power Supply Unit (£39.99 @ Amazon UK)
Total: £518.91
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2018-08-22 15:32 BST+0100
 
The VS550 is junk.

PCPartPicker part list / Price breakdown by merchant

CPU: AMD - Ryzen 3 1200 3.1GHz Quad-Core Processor (£72.79 @ Alza)
Motherboard: Gigabyte - B450M DS3H Micro ATX AM4 Motherboard (£67.20 @ Aria PC)
Memory: Team - Vulcan 8GB (2 x 4GB) DDR4-3000 Memory (£78.27 @ CCL Computers)
Storage: Seagate - BarraCuda 1TB 3.5" 7200RPM Internal Hard Drive (£29.99 @ Aria PC)
Video Card: Asus - GeForce GTX 1060 3GB 3GB Dual Video Card (£179.99 @ Ebuyer)
Case: BitFenix - Nova ATX Mid Tower Case (£23.99 @ Aria PC)
Power Supply: Corsair - CXM (2015) 450W 80+ Bronze Certified Semi-Modular ATX Power Supply (£47.99 @ Amazon UK)
Total: £500.22
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2018-08-22 16:03 BST+0100



 
Depending on what games you play, you will want to adjust the balance of power between the CPU and GPU for best results. I haven't personally used the full gamut of Ryzen CPUs, but it seems to me that a 1200 might be a little weak. If it were me, I'd probably choose a R5 1500X even if I had to reduce the graphics card to a GTX1050Ti, but if your games are very graphically demanding, you may not want to do that, especially since I can always upgrade the CPU later, for less than the cost of a graphics upgrade.
 


My bad, mistakenly pick the vs instead of CX...