It sounds a lot like columnaris to me. Don't buy and treat first until we have the photos but I'll just post some advice in advance if ever the photos should confirm what the exact issue is.
Now, for columnaris, this is a gram-negative motile bacteria. It is one of the most common bacterial infections alongside with hemorrhage septicemia (Aeromonas hydrophila). It will always be present in tanks and will strike when the mucous membrane becomes damaged as a result of parasites inflicting the damage such as protozoans like ich and external parasites like fish lice aside from weakened immune system as a result of the stress generated from the presence of these parasites.
To treat columnaris, you need antibiotics that should hit gram-negative bacteria instead of gram-positive bacteria although a broad spectrum bactericide will work. Be warned that almost all antibiotics are potent and can destroy your beneficial bacteria so watch out for your water parameters in case the ammonia and nitrite begin to elevate dangerously. If possible, treat the affected fish in a hospital tank or tub to save your main tank's beneficial bacteria from the trouble. Get a sponge filter and heater. A hospital tank needs not be decorated for hygienic reasons. It should remain barebottom.
Kanamycin, tri sulfa and tetracycline are often recommended for various bacterial infections. All three can be ingested by the fish if we are dealing with internal bacterial infection otherwise simply dose them in the tank. Follow the instructions carefully labeled on the container. Be very careful not to overdose the antibiotics. Tetracycline will not work effectively if your pH is 7.6 or more.
Another antibiotics I'd recommend are the combination of Maracyn (erythromycin) and Maracyn 2 (minocycline). Both meds are heavy duty treatments and may be lethal if overdosed so again, follow instructions carefully however they should be effective against columnaris.
Acriflavine may work but be very careful not to handle it directly with your bare hands as it can stain. This is best used in a hospital tank, not your main tank. Wear gloves as much as possible when handling it.
There's another which I have not tried yet, Tricide Neo but I do not believe there is a bacterial infection common in the aquarium industry that is resistant to this treatment. It may be worth your while although if this is also lethal, your neon tetras may not stand a chance against this treatment. Neons tend to be sensitive to meds.
For fungal infections (saprolegnia), I'd go with malachite green. A 0.1% solution which is a drop per gallon may be effective against saprolegnia cases assuming the issue is indeed fungal infection although fungal infections are hard to come by and are often misdiagnosed as bacterial infection, a false perception that has already confused countless number of hobbyists of the issue they are dealing with. Only microscopy can accuately help identify such issue.
By the way, welcome to TFK! Please do not hesitate to ask more questions if you still feel the need to ask.