Good afternoon Jonathan Miller,
Welcome to the Apple Support Communities!
I understand you are getting a message "applications" would like to control this computer using accessibility features" after migrating data from an older Mac Pro to your new Mac mini. Could you provide a little more information regarding this issue? Do you have any accessibility apps installed on your Mac? Do you have any third-party apps, like text-expander apps or Dropbox, for example, that have settings to control your Mac via the Accessibility menu?
I would suggest opening the Accessibility settings in System Preferences and remove any apps that you are not familiar with, or grant access to the ones that you use. Click the Apple icon in the upper-left corner of your screen, then click System Preferences > Security & Privacy. Click the lock in the lower-left corner of the window to grant access using your computer user admin account password. Then make the changes that you want.
While you are in System Preferences, go to the Users & Groups pane, and click the Login Items tab. These are applications that start up when you turn your Mac on. Most are "helper" apps or printer apps that generally aren't needed. Make a note of which applications you have listed there, and remove them by clicking the minus (-) sign. (You can add them back later if you wish). When you are finished, click to enable the lock, exit out of System Preferences and restart your computer.
If the pop-up message persists, make a note of what application you are in when you see the message and try to adjust its settings. You can also try turning your computer on in safe mode, and testing in a new user account.
I'll include resources below to help you with these topics. I've highlighted some key information here for you. Click on the links for complete details.
macOS User Guide - Allow accessibility apps to access your Mac
"When a third-party app tries to access and control your Mac through accessibility features, you receive an alert, and you must specifically grant the app access to your Mac in Security & Privacy preferences.
If you’re familiar with an app, you can authorize it by clicking Open System Preferences in the alert, then selecting the checkbox for the app in the Privacy pane. If you’re unfamiliar with an app or you don’t want to give it access to your Mac at that time, click Deny in the alert.
Be cautious and grant access only to apps that you know and trust. If you give apps access to your Mac, you also give them access to your contact, calendar, and other information, and are subject to their terms and privacy policies, and not the Apple Privacy Policy. Be sure to review an app’s terms and privacy policy to understand how it treats and uses your information.
Denied apps are listed in the Accessibility section of the Privacy pane. If you later decide to give a denied app access to your Mac, choose Apple menu 
> System Preferences, click Privacy, click Accessibility, then select the app’s checkbox. To remove an app’s access to your Mac, deselect the checkbox. You can also drag an app into the pane to add it yourself, then select the app’s checkbox."
macOS User Guide - Change Users & Groups Login Items preferences on Mac
macOS User Guide - Set Accessibility preferences on Mac
Use safe mode to isolate issues with your Mac
Isolating an issue by using another user account
Safely open apps on your Mac
Take care.
