Evenhanded means fair to all sides. If your essay is evenhanded, it should look at both sides of an argument, without showing preference for one side or the other.
When you talk about two opposing ideas, you often invoke the idea of hands. “On the one hand,” you begin, discussing the first idea, and then move on to discuss its opposite, starting in with, “On the other hand.” Evenhanded means treating both “hands” evenly. Of course, you're not really talking about hands — you're talking about sides of an arguments, or siblings splitting a cookie.