I’m a first-time cat owner; at the pet store the boss said the mark was just dirt and washing would fix it, but it doesn’t seem to wipe off

From the photo and description, if the black stuff on the cat’s front leg cannot be wiped off and is accompanied by hair loss, scabs, and dandruff, it is very likely ringworm or another skin infection. It is advised to first take the cat for a skin scraping to confirm the cause; at home you can shine a Wood’s lamp on the area to see if there is an apple-green glow, which basically confirms ringworm.

In this case, it is best to do targeted skin care and apply medicine, and put on an Elizabethan collar to avoid the ringworm spreading.