Scanner will not allow files to be saved to desktop

Since you upgraded to Sonoma, you probably also need to update (or reinstall) the HP driver (HP Easy Start, Smart or whatever they are calling it these days). You did say you updated the printer driver, but the scan driver may be separate - you should check that.

You may also need to give Full Disk Access to the HP scan app. Go to System Settings > Privacy & Security > Full Disk Access and add the HP scanning app to the list.

Does anything at > Scan images or documents using a scanner and your Mac - Apple Support help you to see or set the Save location?

Optionally open Image Capture in your Applications Folder, then use and set it to Save the scans to your Desktop.

Image Capture will allow you to save to the Desktop. Is that what you are using?????

I got that you are using the HP scan software, the advice is NOT TO USE THE HP SOFTWARE or check for an update on it on HP's website.

If somehow we are misunderstanding one another, please write in more detail EXACTLY what you mean by "I am using the HP scan function." Please be detailed.

Don't use the HP app, it's poorly written at best as you have proven by showing it has lost functionality. You can check with HP to see if it has updates though.

Most experienced users on these forums use ImageCapture, why.... simple it works! If that does not suit your needs use VueScan from Hamrick.com. You can download an evaluation version of VueScan and if it suits your needs you can buy it.