how to permanently delete jpg file from Time Machine?

Because of the way Time Machine is structured and its intent, there's no way to selectively delete a file. If you delete the file from your internal storage, eventually the file will be disappear as older backups are deleted.

Note that it used to be possible, when Time Machine used HFS+ to format the backup disk. When Apple transitioned to APFS for Time Machine backups, the ability to expunge a single file or folder was eliminated.

If you absolutely, positively need to have the file gone immediately, the only sure way is to wipe your existing TM backup and start over:

1. go to System Settings > General > Time Machine, select the backup(s) with the file and delete them with the (–) button
2. use Disk Utility to format the drive (Erase and reformat a storage device in Disk Utility on Mac - Apple Support)
3. go to System Settings > General > Time Machine, click the (+) button and select the erased disk to start over with a brand new backup

This used to be possible some years ago but it is no longer possible. The Time Machine backup drive is now read-only to the user so you won't be able to do anything to individual files that are backed up. Also, while you may see files in the Finder when you look into Time Machine backup folders, those are actually special links, they are not the physical files themselves. If you somehow find a way to change the contents of a Time Machine backup, it will probably no longer work as a viable backup for restoring other files because those special links will be broken.

If you want to remove that file from the backup, you can erase the backup drive and conduct a new fresh backup after instructing Time Machine in its Settings to skip that file or folder. But this would mean you lose all the backup history in your existing backup.

Christine Altieri wrote:

Sounds complicated. I am afraid I may do something wrong. I'll wait. How long should I wait? Does it go by the number of back ups I perform, or time itself? Thanks.

Is the backup copy of that one jpg file causing some sort of problem? How much space is it using up? Why not just leave it there. It is not possible to remove it explicitly. It might take many months or even years before it disappears from the backup drive through the ordinary slow filling up of the drive and housecleaning that the MacOS does to keep space available for the next incremental backup.

Time Machine won't even start deleting older backups until the available space is full, so if you've got plenty of space on your external drive, you'll have a long wait.

It should not be too difficult to start over, though. Basically, you're unlinking the current backup disk, erasing and reformatting it, then using that same disk for a new backup. You will need to allow some time for the first backup to complete, though.

I am not going to pry but as long as you elected to encrypt that Time Machine backup, its contents are completely inaccessible and useless to anyone who does not know its password.

Choose a backup disk and set encryption options on Mac - Apple Support

Conversely, if you did not elect to encrypt that Time Machine backup, anyone can use it to effectively create a duplicate of the Mac it backed up.