The white stuff is excess mucous produced by the fish to cover the damaged tail.
Make sure you have at least 10 tetras in the group to reduce aggression and stress.
Wipe the inside of the glass down.
Do a 75% water change and gravel clean the substrate every day for a week.
Make sure any new water is free of chlorine/ chloramine before it's added to the tank.
Clean the filter if it hasn't been done in the last 2 weeks. However, if the filter is less than 6 weeks old, do not clean it. Wash the filter materials/ media in a bucket of tank water and re-use them. Tip the bucket of dirty water on the garden/ lawn. Cleaning the filter means less gunk and cleaner water with fewer pathogens.
Increase surface turbulence/ aeration to maximise the dissolved oxygen in the water.
If it hasn't improved in a few days or it gets worse, post more pictures and add some salt.
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SALT
You can add rock salt (often sold as aquarium salt) or swimming pool salt to the aquarium at the dose rate of 1 heaped tablespoon per 20 litres of water. If there is no improvement after 48 hours you can double that dose rate so there is 2 heaped tablespoons of salt per 20 litres.
Keep the salt level like this for at least 2 weeks but no longer than 4 weeks otherwise kidney damage can occur. Kidney damage is more likely to occur in fish from soft water (tetras, Corydoras, angelfish, Bettas & gouramis, loaches) that are exposed to high levels of salt for an extended period of time, and is not an issue with livebearers, rainbowfish or other salt tolerant species.
The salt will not affect the beneficial filter bacteria, fish, plants, shrimp or snails.
After you use salt and the fish have recovered, you do a 10% water change each day for a week using only fresh water that has been dechlorinated. Then do a 20% water change each day for a week. Then you can do bigger water changes after that. This dilutes the salt out of the tank slowly so it doesn't harm the fish.
If you do water changes while using salt, you need to treat the new water with salt before adding it to the tank. This will keep the salt level stable in the tank and minimise stress on the fish.
When you first add salt, add the salt to a small bucket of tank water and dissolve the salt. Then slowly pour the salt water into the tank near the filter outlet. Add the salt over a couple of minutes.
That was my thought but, odd as it may seem, I've never experienced a case of nipping so didn't know for sure. Odd thing is that this just developed today. Out of all the tetras this one has been the most aggressive. Mayhaps the others just got tired of it... 
