Yes you can. I had someone do it for me. He called it "disabling" the corrupted/damaged sectors. Now my old 500GB drive for my old desktop is a 380GB drive that works flawlessly, aside from the storage loss.
There's two methods. The easiest is to run chkdsk from a command prompt and telling it to do a thorough scan.
chkdsk /R e: (where e: is the drive letter)
It'll remap bad sectors that it comes across at the filesystem level. Because HDTune checks the drive surface, you will still see the bad sectors.
Apparently SpinRite can remap beneath that level, so that bad sectors go away. (Until new ones show up) Google it. It's pretty expensive, but many people say it's one of the most useful HDD tools out there.
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