OK, from the demo it looks like each post excerpt does have a class associated with the category included in it. So, here’s a very general rule that you can try:
.fusion-posts-container .category-photography .entry-title:before {
content: url(/images/photo-icon.jpg);
}
You will want to add this rule either through the theme’s Custom CSS option (if it has one), using a child theme, or using a CSS plugin like Jetpack or Custom CSS Manager. You will create one rule for each category that you have. What this rule will do is place the icon just before the excerpt title.
Notice that the middle part of the selector is a class name for the photography category. When you created your categories, you associated a category slug with each category. The category slug is appended to the word category to make up the class name. So you will want to use whatever category slugs you’ve created for your own categories.
The content url, of course, points to the image icon that you want to use for that category. You will have to add some other CSS to style it correctly.