Image Capture Error 9934

This could be a hard drive thing. How is your external hard drive formatted? To get full functionality (and to avoid damaging the Photos Library) an external drive must be formatted in either APFS format or Mac OS Extended (Journaled) format. When you select your external drive and type command-I for information, you can see the Format:

This one is APFS. (Don't use case-sensitive unless you have to!) Additionally, the drive can not have had Time Machine on it since it was formatted.

Does your drive look more or less like this? Let us know.

You didn't describe what method you are using to back up to an external drive. Are you backing up your Photos library by copying and pasting it? Are you coping the photos and attempting to paste or drag them into a Photos library on your external drive? Are you backing up to a Finder folder? Are you using the Image Capture app to transfer the photos to your external drive?

A quick internet search garnered the following from a website:

So, as Richard Taylor said, it may be that your external hard drive is not formatted properly, or perhaps does not have the appropriate settings and permissions.

Also, you can try saving the photos to the Pictures folder on your Mac, and then drag them to your external drive.

-- Rich