Hi howardfromlafayette,
We appreciate you reaching out through Apple Support Communities for help with this message you're seeing on your iPhone and iPad. We'd love to help sort it out.
Does the message appears randomly when using any app or on the Home screen? Or does it only appear when visiting certain websites?
Let's go into Settings > General > Date & Time and turn off the option to "Set Automatically". After about 20 seconds, or if it's already off, please turn it on again, and verify that your time and time zone are accurate. This step ensures your iPhone has accurate time, which may be crucially important when establishing a secure connection online.
If the issue remains and is appearing randomly, it may indicate an issue with one of your online accounts. In that case, you'll want to head into the Settings app on each device, and tap on Mail. Do you have multiple third-party online accounts listed here?
If so, turn off all syncing options for every third-party account (you can skip any iCloud account), then verify that the message "Cannot verify server identity" stops appearing.
If the alert is no longer popping up, you can enable your syncing options again, one account at a time, and testing in between each to see if the issue returns. This will help to isolate the root cause to a specific account or type of account.
At that point, if you're able to identify the root cause, we can recommend deleting the account, then adding it back. This will help refresh mail account settings. Here's a guide with the steps to help, under the section titled, "Remove your email account and set it up again":
If you can’t send email on your iPhone, iPad, or iPod touch - Apple Support
If the issue isn't immediately resolved after this step, we may refer you to the specific account provider. Let us know how it goes, or if you've got any questions whatsoever.
Have a great day!