Can a kitten that is a little over a month old take a bath?
Actually, bathing is fine for cats as long as their bodies can tolerate it.
However, it's generally recommended that owners avoid bathing their cats before they have completed their vaccinations. The main reason for this is that bathing is a significant stressor for cats, which can weaken their immune system and increase their susceptibility to illness. 
Kittens can be vaccinated as early as around 8 weeks of age. The initial vaccination consists of three doses, with each dose spaced 21 days apart. The rabies vaccine needs to be administered when the kitten is three months old.
Then, 15 days after all the vaccines have been administered, you can take your kitten to have its antibodies checked to see if the antibody levels are within acceptable limits. If they are, you can give it a bath. Therefore, it is best to wait until your kitten is around 4 months old before you start giving it a bath.