My Apple mail account on my iMac fails to delete mail and update since installing High Sierra

Does the same problem occur if you restart your Mac in "Safe Mode"?: Use safe mode to isolate issues with your Mac - Apple Support.

Some additional troubleshooting steps to consider:

• Rebuild that Mailbox, or all Mailboxes, according to the following: Rebuild mailboxes - Apple Support.
•  (Apple menu) > System Preferences... > Internet Accounts > select your email service, and de-select Mail. Wait a moment or two, then select it again.
• If all else fails, temporarily remove that Account from that System Preferences pane, and then add it again.

The last two suggestions are described in greater detail here: Set up Mail with email accounts - Apple Support.

Create a backup if you haven't done that already. To learn how to use Time Machine please read How to use Time Machine to back up or restore your Mac - Apple Support. That doesn't have anything to do with your Mail difficulties; it just goes without saying that you should have a backup.

Also, if you are using any non-Apple "anti-virus", "cleaning", "washing" or similarly categorized junk on your Mac, don't. That also just goes without saying.