This question is an exact duplicate of:
Old related post from 8 years ago: Add tags next to the question in list of hot questions
As of now, there's no way to see the tags on a Hot Network question in the sidebar without clicking into it.
We already show the name of the site the question is hosted on when the site icon on the left is hovered over.
Could we add similar hover-text to the title link that will show the tags? Something like:
[feature-request], [tags], [design], [hot-questions], [sidebar]
Especially on sites like Movies & TV, Arqade and SciFi.SE, this would allow users to easily check whether or not the question is about something that interests them without being forced to click in for more context about the title.