Is it permitted to bathe a kid on shabbos if they are super dirty. Covered in mud and dirt etc.
The Ashkenazi minhag is not to take even a cold bathe or shower on the whole body on Shabbos. However since over here there is a need so it is permitted. Technically you can add some warm water to the bath, as long as it will only stay warm, however adding warm water to it from a boiler that is working on Shabbos is not permitted, because it will cause new cold water to enter the boiler and get cooked. Therefore, you can give him a cold bath or shower, or add some hot water from an urn etc.
A few other points. When bathing the child, you would have to wash the child with your hands, and not with a wash cloth or sponge, as this would be sechita. Additionally, you should not use a bar of soap, rather soap that is thin enough to use for washing dishes. Also be careful when drying his hair for sechita, therefore just drape it over his head, and the water will automatically get absorbed in the towel without you squeezing it out.
Shulchan Aruch O:CH 326, and Mishna Berura, Shmiras Shabbos Kehilchso 14, Ohr Ltzion 2- pg. 252