Desktops are not hard to build. They just take time. I would plan for about 1.5-2.5 hours to build one. I would first watch a video on youtube. They are very helpful, and can save some time with a few tips and tricks the maker might give.

For some excellent PC building tutorials, check out youtube channel austinnotduncan. I find his tutorials very informative and clean.

Unless you need a laptop (like my case), get a Desktop. More potential power at a cheaper price. Now if you want a laptop that performs, it's not unusual to spend a lot more than 300 Pounds. Unsure of the exchange, but I bought an excellent laptop for C$1800 (price includes taxes and shipping). Laptops tend to be pricier. Spend the same amount on a Desktop, you have even more power.

For the price, you can consider the Lenovo Z50

If you really want a laptop, My laptop is around that price range and can run pretty much any game at max settings (its not even a gaming laptop but who cares. No one needs those cool leds to play games) but its only sold at turkey (SAMSUNG np350v5c s0etr). I suggest you look at the apples new 12" macbook (Yes. its a joke).

If you can use legos you can build a computer. It isn't difficult as long as you watch some video tutorials. Basically, all you need to do is put each part in its slot and then connect the wires.

I dont really know!

I bought a laptop for £300 and I get 200-300 fps on mc, about 150 on csgo.. It is used though! (near to new when I got it)

Agreeing with what everyone else has said, a desktop would be miles better value. Trust me, one day you'll regret getting a laptop. It happened to me. Even if you don't build it (which I do recommend you do), a desktop is just better value all round

This is pretty much the best thing I found in your price range. I only checked Amazon, so maybe you can find a better deal on another site, but I doubt that you will find a significant improvement. Just take a look at the benchmarks. Pretty much every game is unplayable unless you go on lowest settings at a low resolution. Now, if you build a desktop, it's an entirely different story. This is in your price range and can easily handle games @1080p on medium settings.
PCPartPicker part list / Price breakdown by merchant

CPU: Intel Pentium G3220 3.0GHz Dual-Core Processor (£40.91 @ Ebuyer)
Motherboard: MSI H81M-P33 Micro ATX LGA1150 Motherboard (£32.36 @ Scan.co.uk)
Memory: Crucial 4GB (1 x 4GB) DDR3-1600 Memory (£22.12 @ Amazon UK)
Storage: Seagate Barracuda 1TB 3.5" 7200RPM Internal Hard Drive (£39.59 @ CCL Computers)
Video Card: XFX Radeon R7 260X 2GB Core Edition Video Card (£81.91 @ Scan.co.uk)
Case: Fractal Design Core 1000 MicroATX Mini Tower Case (£21.17 @ Scan.co.uk)
Power Supply: EVGA 500W 80+ Bronze Certified ATX Power Supply (£38.99 @ Novatech)
Operating System: Microsoft Windows 7 Home Premium SP1 (OEM) (64-bit) (£67.95 @ Ebuyer)
Monitor: Hannspree HE225DPB 21.5" Monitor (£81.29 @ Aria PC)
Keyboard: Gigabyte GK-KM6150 Wired Standard Keyboard w/Optical Mouse (£12.12 @ Dabs)
Total: £438.41
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2015-05-03 13:02 BST+0100
Either way, I'd recommend saving up more so that you can get something better. I'd say around £500£-600 would be good.

Depends on what games you want to play. You can't expect from a £400 to play nearly every game out there smoothly (even on low). When it comes to desktops, it changes though. I recommend getting a desktop if the games you want to play are resource intensive. If not, then the laptop is... fine I guess.

Well, for £400 you aren't going to buy a laptop that is 'able to play alot of different games quite smoothly on it'. Sure, you might be able to get one that runs Minecraft at about 40-50fps on a low-medium render distance, but is £400 really worth that? Someone I know has recently made a £110 computer, and it runs Minecraft at far render distance at 50-60fps (with optifine), and it is 4x cheaper than a laptop! So, if you were spending £400 on a computer, you could get something pretty good. Also, if you were worried about the mobility, consider remote desktop connections, you can even get apps on your phone for that.

Exactly what GRH-Gaming said, You can get alot more power into a desktop PC.
It's so much more worth it.

For what you get for your money, a desktop computer would be a lot better. Gaming laptops are always >£400 and are usually a lot more.