Little ducky might have a tiny parasite or fungal infection.
I found this question because of one you posted 5 months ago. Thanks for your friendship. I am 74 and suspect you are close to 15. At your age, i was a farm kid. What people most often have done in your situation is to experiment. Please write down the following and tape it where you will find it from time to time. It is one of my favorite sayings and was given to me by a vetinarian.: "You can not argue with sucess!" "You can not argue with sucess!"
Docktors, all of them, have what is called, "a Practice". Now it is your turn. Just because you pay out money, does not gurantee sucess. I believe you are taking the most intelligent path. I am about to give you some advice from my experiences, which is all a doctor can do, as well.
Try to see what its bill looks like inside the mouth, underneath the bump. If it looks infected, take some paste or salve used to treat athletes foot. One i know is called, Lotrimin AF. Brush Ducky's sore bill with some of your tooth paste and an old discarded tooth brush. Blot it dry with a paper towel and pack the infection with Lotrimin or what ever might be available to you of similar value. As i know, you do not have a lot of choices.
Take a spoon and heat it on your stove top, being careful not to burn yourself or get it too hot. Only 130 degrees Fahrenheit is enough. Hold the spoon with plyers, to protect your self. Rub the top of that bump with the hot spoon to kill the surface infection. Put an antibiotic salve onto the top the treated area. Silvadine is the best for this and could be used in the place of anti-fungal salve as well.
Keep it in a cage, away from mud, with fresh water to drink. Don't feed it for a day. Next day let it eat from your hands, only. We don't want it picking up more germs till it gets well. Keep it in the shade. Set a timer and let it bake in the sun for only an hour a day. The sun has some ultra violet rays that are very effective at clearing up infections. But we don't want it to get too hot, either. If you are able to save its life, or not, you have at least done all you could, within reason.
If you have a dog, consider getting a rooster. They have a way of teaching a dog to behave itself. But you will have to protect it for the first nine months of its life, till it gets big.. My favorite is a Rhode Island Red. But a Game Rooster is very colorful, can fly a little bit and is more agressive.