Corydoras need pure water conditions, so they're not a good choice for cycling a tank. You need hardy fish like Feeder Guppies, Platys, White Clouds or Danios. Fish that don't mind the poorer water conditions. You need several small fish. About 6 should be enough to start the "nitrogen cycle" in your 20 G. It's the fish waste that starts it.
Yo need to get some floating plants into the tank too. Anacharis (Elodea densa) and Pennywort are good.
Once you have the fish and plants in, you test the water every day for traces of ammonia or nitrite and if you have a positive test, you remove and replace about 25 percent of tank with pure, treated tap water. Just test and do the water changes when needed. When you have several days of "0" tests for the above toxins, your tank is cycled.
If you can, return the Corys and get some the fish listed above.
"Fear not, my young apprentice. Just change the tank water."