Save iMac files on flash drive

Have you tried the flash drive on another computer (not necessarily a Mac)? This is a useful diagnostic tool to see if the drive functions at all, even if it is not talking to the first computer. Also, restart the Mac to see if this makes any difference.

Ultimately the drive might be at fault and may require reformatting, but first read the list of tips in Apple's support document If a USB device doesn’t work with Mac and a list at this site which may be more specific to and useful for a flash drive.

It shows you how you might be able to see the drive in System Information and Disk Utility. Sometimes when a USB drive does not mount in Finder it is still seen by the Mac and will be visible in either of these two places.

have you set your finder settings for the flash drive to show up?

go to Apple Menu > Finder > Settings.

and in the "General" tab, ensure that "External disks" is selected for the drive to show up on your desktop.

and then in the "Sidebar" tab, in the "locations" section, ensure "External disks" is selected for it to show up in the finder's sidebar.