Hi Gimmy,
As the listing of your posts isn’t technically an individual page that has been created and published to your site, there isn’t a way to set a featured image on it.
Adding a header to that page would require you to change the theme’s underlying HTML and PHP. If you’re comfortable doing that or would like to experiment then the first step is for you to set up a child theme.
In case you’re unsure, the following guides provide a good introduction to child themes, including steps to set one up:
After you have completed that step, copy the parent’s index.php file to your child theme’s directory and then open it in your favourite text/code editor.
This is the code that defines your your “new posts” page. The simplest way to add an image to the top would be to place the HTML code for it underneath the following:
<main id="main" class="site-main" role="main">
Some good guides for inserting images using HTML are:
Hope that’s helpful!