Hi tech-s and welcome to the Mattermost forums!
What’s the subject and the first few lines of the mail? Is it probably “New Direct Message from @username on $date” and in the body, it says “While you were away, @username sent you a new Direct Message”? If so, the online status detection seems to be broken on your client and Mattermost thinks, you’re away while you aren’t.
Please also check your notification serttings in the client. To do so, click on the gear icon in the top right corner and then on Notifications → Email Notifications. Here’s a screenshot of what it looks like on my end:
What Mattermost server version are you using?