I use a plant called hornwort in all of my tanks, it definitely make a difference with nitrate.
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He was talking Nitrite and not nitrate.
Do not add salt! You have a neon tetra and tetras cannot handle salt as well as other fish do. If you use it, you need to use 1/2 of regular dose.
Going back to the worm infestation, you say you do a water change. That's great but do you vacuum as well? Vacuuming will remove detritus and other wastes that will cause spikes in your water parameters. You need a device that is known as a python or python-like vacuum. (http://www.bigalspets.com/fish/maintenance- equipment/gravel-washer-large.html) Also known as a gravel cleaner.