So sorry to hear that you have got a cold. As far as I know, once you get a cold, you will appear to suffer more eye discharge, some running nose and maybe watery eyes as a result of this. So, it is quite normal for you to see this. But it does not mean that your eyes have some problems. If the situation of your eyes continues to be bad after the cold is recovered, you need to see a doctor.
Well, generally speaking, a cold can cause discharge in the eyes. First, you should know that if you get cold, your immune system can be weaker. And it will make your eyes become fragile. And it is easy for some bacteria invade into your eyes. Your eyes will be infected. And then as we know that when you have eye infection, it eventually lead to discharge in your eyes. Also, eye discharge can be caused by allergies, bacteria, makeup, and chemicals. So I suggest that you schedule an appointment with your primary care physician. He or she can take your temperature, a history of your symptoms, and examine your eye's. If there is any question of whether the infection is bacterial, then you doctor will give you some advices. Of course, you can drink some medicine to get your cold recover first.
Yes, it is possible for you to get the discharge of the eyes in the corner because of the cold. Once you get the cold, your whole body will be decreased at the immune system. You will feel that you will easily get healthy problems. Your eyes may get the infection because of the bacterium. You'd better go to the hospital to have a check on the eyes.