I have a protruding vein directly under my right eye that has become painful. What treatment would you recommend?

There are many options for the treatment of veins around the eye (periorbital) including laser, removal through small incisions (microphlebectomy), sclerotherapy, ohmic thermolysis and micro needle radio frequency.  What is done is determined by the size of the vein, amount of veins and their location.  Sometimes combinations of treatment may be advised.   Because of your recent onset and eye discharge, the cause of this should first be evaluated.  The swollen eye veins may be part of an allergic on infection reaction.

The treatment for varicose veins under the eye was recently profiled on National TV by Dr Raffy. Here is the link below