My Western Digital External HDD stopped working. It is formatted to NTFS. If I plug it in, the light on it is blinking, you can hear it spinning, but it is not recognised in Windows, and in the partition manager it is not listed.⏎

If I plug it into a Mac, it is recognised but very slowly, I can see the file system and copy certain files, but many files fail to be copied. But even the copying is extermely slow.⏎
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Sometimes when I plug the drive into computers it is only beeping repeatedly (not clicking), I need to move the USB cable a bit at the connector of the drive to stop this. ⏎
Is is possible to repair this drive at home?

Possible, but if the data is very important to you its best to seek a pro doing it.

From your description it sounds like a possible firmware fault or a bad electrical contact.