Any hope for Imac that wont boot because of missing EFI file?

First post here, a bit rambling but dont have all the facts at hand right now. Have an older Imac, FusionDrive, 21,5" with the glued screen, one morning waking it up from sleep I briefly got the desktop and email notifications, then it shut down. Starting up again it would not boot, it troes but progress bar stops at some point and it does not go any further. Did some searches and eventually got a message on screen saying something like "cannot boot because of missing EFI file". I have tried just about everything, safe mode, recovery mode, Internet recovery (computer does not have this I think) , resetting all kinds of things, no luck. Tried creating a bootable usb disk on my MBPPro but that does not help either. I can attach my Time Machine and get that up on the screen at startup as a choice, same as the bootable usb disk but it does not make a difference.
See some people have the same problem and I have not found anyone that managed to solve this.....
Took the machine to an authorized Apple repair facility in town and they did a check, said it probably was the motherboard but they refused to touch the machine since it was too old and obsolete....