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To glue a wound shut, it's not necessary that it be dry. In fact, SuperGlue works a bit better if the edges are wet, as the goal here is to weld living tissue together so that it will mend. For that, you want clean fresh (i.e. wet and bleeding) edges.
To begin with, flush all dirt and grime out of the wound with fresh water in a squeeze bottle. Once the wound is clean and moist, pull or push the wound closed while you "spot weld" the edges together with SuperGlue. You do not want to put the glue inside the wound -- you are closing up the top, not putting in d