mouse does not work on opening screen until ten seconds later

Also, it helps us quite a bit if you tell us the Model Year of the iMac and the current version of Mac OS installed. iMacs have been available for well over 20 years and Mac OS has too, so we have zero idea of what you own. This is critical information when we are attempting to troubleshoot your problem.

If you simply do not know the model year and version of Mac OS installed, please click the Apple symbol in the upper left of the display and then click "About this Mac" and that information will show up.

Have you run Disk Utility's Disk First Aid on all connected drives?

In Disk Utility>View, select Show all Devices, highlight the top left entry.

Run Disk First Aid on all items in the left panel, from top down.

Are you running any VPN, Anti-Virus, or Cleaning apps?

We need to see what all is running, a report from this will not display any personal info...

Using EtreCheck - Apple Community

EtreCheck is a FREE simple little diagnostic tool to display the important details of your system configuration and allow you to copy that information to the Clipboard. It is meant to be used with Apple Support Communities to help people help you with your Mac. It will not display any personal info.

https://www.etrecheck.com/

Thanks for Old Toad’s etrecheck instructions…

Slow iMac 2017 - Apple Community

Use the Note tool on the bottom of this editor's toolbar, as shown in the image, to copy and paste the output from EtreCheck. In a Reply before you click post, look for this to add longer texts...

Hi,

Is it a wired or wireless mouse? If it's a wireless, I suppose it takes time to pair the mouse.

Do you charge the battery of mouse lately?