I was actually just doing this myself yesterday! Yes, you can add images to the widget content through the HTML text editor (see screenshot)
You can use the <img> tag to add the image to your widget content. If you're new to HTML, you can read more about it on W3 Schools website here.
Here is an example of the format from a widget I configured:
<img alt="Widget Icon" height="50" src="http://angler.maps.arcgis.com/sharing/rest/content/items/2f6de3e17c60448380faa9c375e01905/data" width="50" />
To use your own screenshots, you will need to add them to your content or some other image hosting service to be able to get a URL.
Hope that helps!