Including Category in Preview

For each post I would like to always include the category within each preview boxes in the grid archive of the main page. I saw the thread about always displaying the preview title within the archive grid (below). The theme editor setting is only for “Show Preview Title” and I would like preview boxes to show the category above the post title. For example, my first post should read Books (and then the title of the specific book which is the title of the specific post), the next three posts are Talks, my next two are Essays, and the next two are Reviews, etc. Above the post title in a slightly smaller font in the green theme I would like each post preview (the actual titles of my essays, talks, books, etc.) to also say what category (Books, Talks, Essays, Reviews, etc.) it is. Whatever help you could offer would be great!

https://www.ads-software.com/support/topic/no-hovering-post-title-featured-image-to-stay-visible/

The page I need help with: [log in to see the link]