There's no connection. "Ash" means whatever is left after something burned: "The logs in the campfire burned until nothing but ashes were left." "Potash" can be an outdated word for the element potassium, symbol K, atomic number 19, https://en.wikipedia.org/wiki/Potassium . It is usually a term for mines and mining companies--vague word for a number of different minerals that are potassium salts, and are mined for various industrial uses. (It can also be a surname! It's not common but I do know some people in the United States whose last name is "Potash!")

"Potash" is a specific mineral mined from the earth for use in industrial applications. "Ash" is just the material remaining after a fire has burned out, particularly a fire that burned organic materials like plants. The origin of the world "potash" comes from "ash", but in the modern world they have nothing to do with each other. Unless you are involved in the chemical industry, you will have almost no reason to ever use "potash."

Apparently, the word "potassium" comes from the word "potash". Anyway, "potash" is a specific kind of salts that contain potassium. It is used for fertilizer. I had never heard of it, so I had to look it up on Wikipedia. "Ash", on the other hand, is a very common word. It is what is left from something after it is burned. For example, "cigarette ash".