Not all files are showing up in Target Disk Mode

I’ve no direct experience with Target Disk Mode but I make the following suggestion.

• On the primary Mac create a new folder on the desktop, copy the files in question to it and see if the secondary Mac can see these files.
• If yes see if you can copy them over.

It helps to know the exact model of the older Mac and the OS used on it. Having the same information for the newer host Mac may also be useful as well. You can get this information by clicking the Apple menu and selecting "About This Mac" and by entering the system serial number for the non-booting Mac here:

Check Your Service and Support Coverage - Apple Support

Where are the files or folders located which are missing? In the home user folder or outside of the Users folder?

Did you have multiple user accounts on the old Mac? Or even on the newer host Mac?

Was iCloud enabled? If so, then it may alter the behavior of those specific folders. I don't use iCloud so I am not certain.

There is also a chance of a file system issue or even a failing drive. Try running DriveDx to check the health of the older iMac's hard drive. If connected by USB, then you will need to install a special USB driver. However, even with the special USB driver it is not always possible to access the drive's health information. Post the complete text report here using the "Additional Text" icon which looks like a piece of paper. If the old iMac has a Fusion Drive setup, then post a report for both the SSD & HD.

If the drive(s) are healthy, then try running Disk Utility First Aid on the older iMac's volume.

Maybe try using Carbon Copy Cloner (CCC) to transfer the items from the old computer to see if CCC can see those missing items.

The hard drive is failing with over 4K sectors already reallocated. The only "good" thing is currently there are no "Pending Sectors" which are awaiting reallocation. If there were sectors pending reallocation, then you would likely be seeing slowness trying to access certain areas of the drive. Of course once you start having this many bad sectors on a hard drive, they usually just keep accumulating until the drive won't respond. The next time you try to access a physical area of the drive near the damaged section you may start to see issues and an increase in the "Pending Sector" raw value.

Since there were over 14K uncorrectable errors, you definitely have some data loss if some items are not showing up.

You may want to use CCC to transfer your files if you still have stuff to transfer. CCC has an option to ignore errors so maybe it will allow you to transfer the remaining items. However, if the number of "Pending Sectors" or "Reallocated Sector Count" increases or CCC encounters too many errors, then you may be putting the remaining data at risk attempting to use standard apps to transfer the files as they are not capable of dealing with the errors being produced (nor can macOS). Normally in this situation I would never advise to use CCC for this purpose, but as long as CCC only has a small number of files it has trouble reading, then it may not be too risky (I've used CCC for this when less than a dozen files had issues transferring). I'm only suggesting CCC here because currently there are no "Pending Sectors" waiting reallocation, but if you see any listed I would stop as you will likely be making the failure worse where even a professional data recovery service may not be able to help. The "Pending Sector" raw value gets reset to zero after the sectors have been reallocated so it is important to be careful.

If you still have data to transfer, I highly recommend you contact a professional data recovery service such as Drive Savers or Ontrack since you will likely only get one chance to save the data. Choose wisely. Both vendors provide free estimates and both are recommended by Apple and other OEMs.