As I search for the way to check my email on the admin dashboard so I can quickly check my messages without needing a separate tab.
There wasn’t any that was available to use with the lasted versions of WordPress and when I did come across a plugin, it’s already out of date and no one will support it.
There’s a simple solution that only requires some HTML and access to the hosted site if any.
To pull this off you’ll need to get:

• Custom Dashboard Widget plugin
• Rainloop Web-mail Client (or any other if you prefer)
• Filezilla or any other ftp app (skip this if its already on a server or outside your domain)

Setting up Email Client

We’ll use Rainloop Community Edition to For this post.  (Skip this and go for the Plugin instructions if your using a different one)
Download the zip files and extract the contents to a single folder or a place you’ll known where to find.
With an ftp app like filezilla or the builtin one from your web host if any, upload the folder to your domains main directory or if using a sub domain put the files outside the folder and upload the index files instead.
It’ll go by public_html or something similar.
To access the web client, use your web browser and type either www.yourdomainslink.com/whatever-your-folders-name-with the -files-are-in or if your using a sub Directory just type that in and you’ll see a login screen if successful.

Depending what version of rainloop your using, here are the configuration instructions in case this post can’t be up to date and go from there. https://www.rainloop.net/docs/configuration/

Setting up the Plugin

Assuming you installed the plugin. you go to either to the dashboard Sidebar to Settings, then Costume Dashboard Widget or Configure when you hover around on the right top corner on the widget itself.
Insert the following on Widget Custom Content:

<iframe src=” https://www.yourdomainslink.com/?/MobileVersion/” width=”100%” height=”500px”></iframe>

Just replace yourdomainslink.com with the given URL or sub directory containing the index files.
Change the title of the Widget Custom Title to anything else (Optional)

Final Conclusion

It may required to log in but this should do the job. you may end up retyping the user email and password on separate devices or browsers.
Hope this little work around helps [^_^]/