This sounds like a corneal lesion and the red dilated blood vessel is what is normally seen in such conditions. If you use contact lenses stop them till the lesion heals. You need an ophthalmologist to look at the corneal problem and decide whether it may be infective, following a foreign body in the eye or inflammatory in nature. Another condition that could be what you describe is a pterygium
which is a degenerative condition that is not really an emergency. Have it looked at since it has been there for a month and there is no improvement.