I have bought a cheap Burwood childrens (?) electric guitar, but it goes out of tune VERY easily (all the time!); the tension in the strings seems to be way too slack when tuned in same way as normal length electric guitar. Does it need much heavier strings to compensate? Or is it just a fact of life for such small (and cheaply-made) guitars?
Shorter scale guitars (typically made for children) are easier to keep in tune if you use a higher tuning (such as A - D - G - C - E - A).
The scale length is the distance between the nut and the bridge or saddle which is the part of the string that vibrates. The shorter the length between nut and saddle, the less tension required to reach a given pitch.
Using heavier gauge strings will increase the amount of tension and help. But you may also want to consider tuning to A for half size and G for 3/4 size.
To tune to A starting with the fattest string (normally low E and technically the 6th string) use a chromatic digital tuning device (or phone ap) and tune A - D - G - C - E - A
Or try tuning to G for less string tension but still more than if in E.
Fattest string G then C - F - A# - D - G.
A similar question on Stack Exchange will provide more information that you should find helpful.
Click here to see Short Scale Guitar Tuning
The 'half size' is probably a misnomer. Rather like 1/2 size violins, which are bigger than 1/2 size. Given that a standard guitar spoken string length is around 25", it's an easy sum to do, to arrive at string gauges that will work. An 'ordinary' set of tens uses, well, .010 for its top. If your guitar is, say, 2/3 as long, then a 013/ 014 would be about right, if my sums are correct! HOWEVER - it may well be the naff construction that is making it untune, and putting even more strain on the neck could be a death knell for it. Back to the shop sounds like a good move!
I think that you may be misunderstanding the idea behind half sized guitars. They are simply made for children. The scale length should remain the same in proportion to the size of the guitar.
You would likely not have a reduced scale length guitar in the manner which is usually referred to among guitarists. The strings should not have all that much effect on a good half sized guitar there is just a whole less of it used for the smaller instrument.
That being said it should probably see some sort of professional as there could be a myriad of intonation problems that may be the heart of the problem.
FWIW, there are 3/4 strings available. I found them at Lindo
Using higher tuning is not smart...how can the child then learn to play any tunes? Even if they can they won't then be able to reproduce them on a correctly tuned guitar. There are no special strings for a shorter scale guitar. If you're going out of tune, first be sure that the strings are not slipping on the tuning posts, this often happens with nylon strings and in extreme cases you may have to tie them off. Regarding my first comment about the tuning, if you believe that higher tension is curing your string slip (it isn't) then buy high-tension strings instead.