yes.
https://www.easeus.com/resource/hard-disk-data-recovery.htm
Since your hard drive is functional, it's the partition formatting that seems bad. Not exactly sure.
However, software like what I linked doesn't care. It simply scans ever part of the spinning platters and reconstructs the files. It doesn't care if the file tables are intact.
Once you generate a list you copy to another drive preferably. (it will take HOURS to scan)
Not sure if there is good, free software for this.
OTHER:
Not sure if the HDD is failing or not, though I suggest you do this:
1) obtain another HDD of equal or larger capacity
2) copy everything over on the working partitions
3) run EASEUS data recovery wizard (or other) to reconstruct your raw data files (documents, video etc) but not progams that are installed
4) copy that data to the other drive
then
5) FORMAT (FULL, NTFS), then
6) run HDD diagnostics
alternatively)
7) some weird issues (if still errors after format) can be fixed by a LLF (Low Level Format) http://hddguru.com/software/HDD-LLF-Low-Level-Format-Tool/
(use as free version.), then
8) repeat FULL NTFS (hours) and retest with HDD diagnostics (from HDD manufacturer... includes Drive Short Test to start.. if DST fails drive is bad though)
OTHER:
*You'll need to get a USB, 2.5" HDD adapter likely for all this to work with a laptop, or alternatively REMOVE and put into a computer case (don't mount, just attach the SATA data and power cables, use as secondary drive. don't boot to it. then copy data to that PC...
If you can copy data over, maybe you can DELETE the partition (in Disk Management) then add back the partition. Retest.