Be sure your hands are clean in order to avoid introducing any bacteria into an open cut. Wash with clean, running water and scrub your hands with soap for a least 20 seconds (sing "Happy Birthday" twice). Then, rinse well and dry your hands on a clean towel. If the cut or sore is bleeding and is very near the edge of the nose, then gently apply pressure, using clean materials, until the bleeding stops. Do not block your breathing, and do not pack the nostril. If the injury is not clearly visible or is not right on the edge of your nose, then use the following first aid methods to stop the bleeding:  Sit up straight and lean forward. Maintaining this position helps to reduce the pressure in the vessels located in your nose and prevents you from swallowing any blood. Pinch your nose shut, using your thumb and index finger, and hold it shut for about 10 minutes. Breathe through your mouth during this time. After 10 minutes, release the hold. If your nose is still bleeding, then repeat the procedure. If it still bleeds after 20 minutes, seek medical advice, since this might be a sign of a more serious injury. Stay cool during this process by wiping your face with a cold cloth or sucking on something cold, like ice chips. To reduce infection and possible complications, you can use sterilized tweezers to remove any debris that remains in the cut. Take care not to aggravate the injury or cut yourself with the tweezers. If you think something is lodged in the area, or if you just need to clean out any skin fragments, tissue, or blood clots, sterilize the items you plan to use. If you are not able to sterilize the tools, make sure they are as clean as possible. To sterilize your tools:  Wash your hands thoroughly with soap and water. Wash any utensils, like tweezers, thoroughly with soap and water, then rinse them completely. Place the items in a pot or pan with enough water inside to cover every item. Cover the pan with a lid and bring the water to a boil. Continue to boil the water for 15 minutes with the lid in place. Remove the pan from the heat, keeping the lid in place, and let it cool to room temperature. Drain the water from the pan without touching the sterilized items. If you are not ready to use the items, then leave them in the drained pot or pan with the lid on. Carefully remove the items when you are ready to use them. Avoid touching the parts of the tools that will come into contact with the injury. Touch only the handles or grips. Usually, using soap and water is the best way to clean a wound, cut, or minor injury to the skin. In some more delicate and sensitive areas, products that are both cleansers and antibacterial agents are sometimes recommended.  One common product that is both a soap cleanser and anti-infective is called chlorhexidine. It is available without a prescription in most retail pharmacies. Chlorhexidine should be highly diluted before using on mucous membranes, such as the inside of your nose. Read the product labels. Do not use any product that is not approved for use inside the nose. In order to reach the cut to clean it, you may need to carefully use a cotton swab or a piece of rolled up gauze. Use fresh water and mild soap or a small amount of chlorhexidine on the end of the cotton swab or gauze. Repeat your method with clean, fresh water and clean tools to rinse away any soap residue. Use clean or sterilized tweezers to hold the gauze in order to effectively clean the area. If you cannot easily see the cut, or reach it, then you may have trouble properly treating the area. You could do additional damage or introduce bacteria if the cut is up inside the nose. Call your doctor or go to an urgent care clinic instead of trying to deal with the cut yourself.

Summary:
Wash your hands. Stop the bleeding with gentle pressure. Remove any debris from the cut carefully. Use clean tools to clean the area. Choose your cleaning agent. Clean the tissue around the cut. Seek medical help if the area is hard to reach.