How sound is the belief in the prohibition of washing clothes on Tuesdays and Wednesdays, based on what is rumored that Allah created the detestable on Tuesday, or that whoever does so on Wednesday will be exposed to people's accusations? And how does one respond to those who cling to such superstitions in their daily lives?

Yes, it is stated in an authentic hadith that Allah created the disliked on Tuesday. However, this does not necessitate prohibiting anything on Tuesday, and there is no harm in washing clothes on this day or any other day. There is no legal evidence prohibiting washing clothes on it, and we do not know of any scholar who has said so. The mother must fear Allah and refrain from issuing fatwas, for prohibition belongs to the Sharia, and prohibiting what Allah Almighty has not prohibited is a fundamental principle of misguidance.