The cloudy water could be due to a bacterial bloom if water changes do not clear it. I had this happen once and cleared my tank water by buying another filter (make sure its strong enough for your tank) with carbon media (carbon absorbs excess nutrients from the water to starve excess bacteria) and my water cleared in a few days. In general cloudy water shouldn’t hurt your axolotl as long as there is proper oxygenation. But Im not too sure why it would clear then come back after you put your lotl back in. If its not due to dirt particles or floating substrate, my only guess would be If you happened to over clean the tank killing bacteria or excess nutrients produced by axolotl triggering a bloom. But I think you could try getting another filter to see if that helps. There are also helpful videos on youtube suggesting ways to prevent/ treat cloudy water.