The biggest problem is that regular CA glue (super glue, crazy glue, etc.) dries rock hard and tends to crack and peel off. Also, don't put the glue in a cut, put it over a cut. You need to be able to close the wound completely before putting the glue on top. If you put the glue inside the cut, your skin cannot heal back together because the glue will act as a barrier, making it heal more slowly. Deep cuts may also need to drain and superglue will seal the skin off completely. It will most likely burn like holy hell too. The liquid stitches glue they use in doctor's offices (derma-bond) is also CA based, but stays flexible, acting like a second skin. I have tried a number of over the counter products (liquid bandage, new skin, etc.) and none of them work very well. You are better off ordering actual derma-bond online. Medical supply stores usually have it. Try Amazon too.
If you have a hobby shop nearby, they often have flexible CA glue for building model airplanes and such (Insta-flex is one) that works much better than regular super glue.