The smudge you're seeing is from the mail sorting machines. Here's a picture of ours as well (we send one to ourselves). If you look at any mail piece, you'll notice they all have a smudge around the same area on the opposite side of the address. Their machines are pretty rough. We put on a UV coating to try to help and it would be much worse without it.

The only way to protect it 100% is to laminate the cards but this would double the price of the postcards due to the labor intensive nature of the product. The main way to address this would be to design around that particular are of the postcard so that no critical text hits that area.