Red Splotches seems to happen in adults as well as with children. This is called a sub-conjunctival haemorrhage. It means that a microscopic blood vessel in the white of the eye has ruptured, and blood oozes out in the space immediately below the surface. It may be quite extensive and can look alarming, especially if large.
In most cases this condition will clear itself up with no therapy. After a few days it will tend to darken into a brown colour. This will fade into a yellow and finally disappear as nature resolves it entirely.
Any recurrence should receive medical attention. In some instances, albeit rarely, important underlying conditions may be the cause. This is usually unlikely – but I don’t believe in taking undue risks, and like to double-check most things, as you know.