Both of the above are good builds. The Pentium G3258 needs to be overclocked to achieve decent performance. The same could be said of the APU build, which would be my choice for a kid's build at this point.
I believe (but am not sure) that Intel had OEM's disable H81 overclocking for the G3258 CPU quite some time ago. Whether or not they backtracked on that I do not know, but buyer beware and definitely do research if you plan to go that route. A nice alternative to the G3258 system would be an Athlon 860 based system which is essentially an A10 APU with no GPU cores available. Pair that with the RX460. That will give you better performance for about the same price vs the APU option so that might not be a bad way to go. Again, you want to overclock the Athlon. Pretty much any CPU you're going to get into at this price point requires overclocking to achieve "desirable" (depending on who's desires we're talking about) performance.
Nobody has asked btw-- does 300 GBP include the Windows license or do you have a source for that? If you need a machine for 300 WITH windows you're going to get a scum box.