It may hurt, but...
There is a small piece of nail still embedded in the corner. You need to get that out.
Soak to soften the area and try to get it free. If you can't get it out on your own, a podiatrist can. Sometimes this can be done without numbing the toe, but often, anesthesia
is necessary. If it is a chronic condition, there is a very successful surgery that can be done to permanently correct the problem.