I'm talking about something like a backup copy located on a drive, in addition to the main.
If there is, how would I be able to retrieve it and write the backup back to the main partition?
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The MBR partition table does not have a backup, but some tools can try to discover old partition locations by scanning the whole drive.
(A GPT partition table does have a copy; the backup GPT header is on the very last sector of a disk, while the partition entries precede it. On Linux,
gdisk aka gptfdisk can restore the backup to main GPT. Many recent computers in fact use GPT+UEFI, rather than MBR+BIOS.)
The MFT "master file table" is a filesystem concept, so it is per-partition and depends on which filesystem each partition uses. For example, NTFS does have a backup copy, which TestDisk can restore files from.