Corrupted HDD
A proper format of the HDD with problems will absolutely prevent recovery of data from it, as the data will be gone. A format of the drive = data gone.
Proper format (the one that takes a while) implies writing data over each sector, and reading back the data to make sure the data was properly written) so if you format it then you basically are unlikely to recover anything.
Quick format just erases the file allocation tables, basically the area where all the information about the files was stored, so a recovery software would be able to scan the surface of the drive and recover data based on various signatures (ex. a zip file would have the sequence "PK" at the beginning, a PNG file would have PNG in the file at the start and so on)
If you have the new SSD, I'd suggest reinstalling Windows on it, then make a folder on in and copy the contents of the hard drive onto the SSD and then mess around with the drive.
I'd suggest the WD tool WD Data Lifeguard Diagnostic (it works with any mechanical drive) : https://support.wdc.com/downloads.aspx?p=3&lang=en
Select your mechanical drive and use the function erase / wipe disk which will overwrite all surface of the drive just like a full format but is faster, and makes the drives re-test bad sectors or sectors that may be on the edge of going bad and mark those sectors bad.
After that process and after a reboot, the SMART data should be updated with the proper number of bad sectors or reallocated sectors.
Then, you could use various tools to see where those bad sectors are located and if they're all located in a specific area, you could create one or several partitions on the drive in such a way that the area with bad sectors will be outside any usable partition, so you won't risk losing more data by writing stuff near that area which may be a weak spot in the disc surface inside the drive.
For example, if you have multiple bad sectors or reallocated sectors at 72-73% of the drive surface, you could make a partition from 0% to 70% and give it a letter, one partition from 70% to 75% which you leave without a letter , and a partition from 75% to 100% of the drive which you give a letter. You lose 5% of the drive capacity but you reduce the risk of that damage expanding or losing more data, because now you're never writing or reading files from that area which is weaker.

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