For that HDD, I'd recommend keeping that, and just buying a smaller drive (like a 128GB) just for your boot drive, because trust me, if you're gonna be playing games, you'll use up a lot of space quickly.
I have got a couple issues with that build though, for one, the CPU is worse then the one I recommended, the i5-6600K is better then the i5-7400 in pretty much everything, the 7400 is just a bit cheaper, but not by very much. If you're wanting a newer processor that's better then the 6600K, then go for the 7600K, but even that for the extra £20 price tag, isn't that much better then the 6600K. (
http://cpu.userbenchmark.com/Compare/Intel-Core-i5-6600K-vs-Intel-Core-i5-7600K/3503vs3885 (With the core that's better then both the core in the $700 build and the core I recommend.) -
http://cpu.userbenchmark.com/Compare/Intel-Core-i5-6600K-vs-Intel-Core-i5-7400/3503vs3886 (With the core in the Under $700 build)).
The GPU though, get the 1060 if you can though, since yes, it is a better card overall, I just recommended the 1050 Ti since, well, budget ¯\_(ツ)_/¯
That build hasn't got a cooler though (Or well, it maybe has a stock one, but they're not the best things ever), so you'll have to get one yourself. Again though, I'd recommend the cooler I suggested above.
With the $800 build too, the CPU isn't as good as the original CPU I recommended (and still not really any cooler).