Adding Images to category and how to display in category widget as featured

Hi there,

it would require Custom Development.
1. WP doesn't natively support Featured Images for Categories. There are plugins that enable that feature but all they do is provide the function to add images to a taxonomy term.
2. The WP core Category Widget does not include any filters - so it cannot be updated to display said image.

The alternative is to use an Image Widget and create your own list of categories.