Nitrates-help please

So my tank has been doing wonderful when all of a sudden I had a major spike in Nitrates and Nitrites-ammonia is zero. Only changes to the tank that I have made, aside from water changes is switching my Axolotls from blackworms to earthworms, plus added duckweed and pond lettuce-that was added two weeks ago. I removed the duckweed and pond lettuce, did two 25% water changes yesterday and today Nitrites are at zero but the fricken Nitrates are still showing 20 ppm. I tested my normal tap water-city water and that is testing at 10 ppm. So first of all how is it possible to have Nitrites test zero but Nitrates still show up-and is it possible that the water test is bad seeing that it is showing Nitrates from the tap water. I don't know what to do, is there anything I can add to just remove the Nitrates or at least lower them-would buying bottled water be a better solution?
I do let the water sit for 24hrs before I add it to the tank. I am so worried about my poor babies right now!

Nitrates with an A are the final product of the nitrogen cycle - seeing them is actually good! They're acceptable up to 40 ppm in freshwater tanks. A level of 20 ppm is fine. Nitrates mean your tank is cycled!

They can be removed via weekly water changes.

Thank you very much for your reply Kaini, I was worried once I saw them climbing again! Guess I better first learn not to panic and second brush up on tank cycling again.