I have/had something similar to what you are describing, End of my nose got red and was sore. Eventually, a small area formed a head and was draining liquid/pus and some blood on expression. I mentioned it to an ENT doc at an appt and he said it was probably an infected hair inside my nose and told me to put Neosporin on it and inside the tip. I did this and it seemed to take care of it. I mentioned this to another ENT doc and he prescribed Bactraban (?) and told me to put it on the outside and inside my nose. I haven't used that yet. Recently, it was getting red and a bit sore again, just the end. I started using the Neosporin like before and it settled down. You might try this but if it continues, I would have it checked and good luck finding a good doctor anymore.
Please have it seen by a doctor.
It could be anything from a simple scratch that keeps re-infecting to something much worse like cancer.
Any time a sore doesn't heal it should be looked at.