Yes, you can do exercise with that. The broken vessel in the white of the eye is called subconjunctival hemorrhage. You could not realize it by yourself unless you see it in the mirror because it is painless. It is can be caused by some heavy activities, including coughing, sneezing, vomiting. It is a common condition because the conjunctiva could not absorb the blood effectively, but it would disappear within two weeks. Hence, you can do some exercise with that condition, but you had better not do any strenuous exercise, otherwise there would have more blood on the surface of your eyes.
You had better not do any exercise in recent days. Subconjunctival hemorrhage can be ascribed to fracture of the vessel of ocular conjunctiva of itself or the fracture caused by the external force due to the frail blood vessel of ocular conjunctiva. The causes of hemorrhage include trauma, pressing of neck, violent cough, conjunctiva inflammation, ateriosclerosis, high blood pressure, nephritis, diabetes mellitus, bleeder disease, helopyra, exogenous febrile disease or cholera. Generally, sufferers do not need to receive a treatment and it can disappear naturally about a week later. But if the hemorrhage occurs frequently, you need to watch out and go to hospital for an overall examination so as to check out whether you suffer other diseases such as high blood pressure, diabetes mellitus, etc. and help you receive a timely treatment.
Well, as far as I know, broken blood vessel in the eyes is a sign of your hazardous health conditions, especially your eyes, as well as other parts. Anyway, this is quite a serious thing, therefore, it would be wise for you to get adequate rest to recover and get some treatment. Of course not right now since it would make the situation even worse. Maybe a few days later you are good to go, as long as you pay due attention to your eyes.