Hi,
There is no option for that by default in HikaShop.
You could create a custom field of the table "category" and of the type "ajax image" so that you could add badge images to categories.
Then, with a bit of PHP code in a view override and a bit of CSS to place the image from the custom field on top of the category image on the listing, it should be doable. So with basic PHP/CSS skills, it should be possible.
Now, to have something more generic, like for example the possible to display several badges on a category, the possibility to have restrictions, like only display the badge if all the products in the category are out of stock, etc, that would be way more complex.