Nitrite spike in cycled tank

My 20 gallon tank finished cycling about a month ago. Since then I have added three ~19 cm axolotls, one per week. The last axolotl was added a week ago. I've been doing weekly 20% water changes and checking the water chemistry at least weekly. Everything has been fine until yesterday, when I noticed the nitrites going up. Ammonia was 0.25, nitrite 2.0 and nitrate 5-10 ppm. I did a 20% water change and cleaned the hang on back filter with dechlorinated water. The filter was pretty dirty and the intake tube might have been partially clogged (I couldn't tell just by looking in the tank; otherwise I certainly would have taken care of it sooner). I also noticed the flow rate was pretty low (I previously couldn't tell the flow rate was low because I placed a large filter sponge at the filter outtake to reduce the flow into the tank). I have since turned the flow rate up. Today the ammonia and nitrite are the same as yesterday but the nitrate has increased to 40ppm. I will do daily water changes until the parameters are stable again, but I'm worried because I will be going out of town this weekend. There will be no one to do a water change on Saturday.

A few questions:
In the short term, will the axies be ok missing a day's water change, assuming the water parameters do not improve before then?

In the long term, do you think my water parameters will improve after filter cleaning and increasing the flow rate? Should I add more biomax pellets to the filter to support a larger colony of bacteria? Do I just have too many axies in the tank? I'm planning to keep 2 and rehome the 3rd. I am feeding mostly earthworms but also sometimes pellets. I have play sand as a substrate so it's difficult to find any uneaten pellets in the tank. I know the pellets were really messy when I still had the axies in take out containers. Could too many uneaten pellets have caused an ammonia spike?

I have had these little guys since they hatched and I am trying to take good care of them. Thanks in advance for any help!

Ammonia and nitrite readings in a cycled tank are definitely a sign of inadequate filtration. You have a heavy bio load in the tank with three large axis. Any of the options you mentioned will help. I would try adding additional filtration first. Your filter should have as much biological filtration media as it will hold. You could also add a sponge filter to help manage the waste levels. A seasoned sponge filter would be the quickest fix, but any sponge filter will help long term once the beneficial bacteria have time to colonize. Ideally, you should upgrade to a larger tank for three large axis. It is much easier to maintain stable water parameters in a greater volume of water. In the meantime, regular water changes and improved filtration should help manage the waste. Cleaning up any uneaten food will also help. Test frequently to ensure the additional filtration is adequate to manage the problem. Hope that helps.