Before declaring your external HDD is corrupted, you can re-plug your external HDD to another USB port or try another USB cable to fix this issue first.
A faulty connection, though it is foolproof, might be the key fix if the external HDD is not mounting. If you are using a USB connector or adaptor, change to another one to connect the drive to your Mac.
You are able to try a super slow jam. Some users found that their external HDD will mount properly when they plug it in exceptionally slow motion. However, if they try and plug in the external drive normally, the Mac doesn't recognize the disk.
You can check the Finder Preferences. Open Finder > choose Preference > check External disks both in the General tab and in the Sidebar tab to show your external hard drive on your Mac computer.
If your Mac could not mount the external hard drive on the desktop still, let's troubleshoot this issue with the following solutions.
You're allowed to force mount the external HDD in Disk Utility. Launch Disk Utility > click on the Show All Devices option in the Disk Utility's View menu to show the external hard drive icon > choose the grayed external drive in the Disk Utility's sidebar > select Mount in the top center of Disk Utility.
Or, you can repair this unmountable external hard drive with First Aid. Open Disk Utility on your Mac computer > select the unmountable external hard drive on the left sidebar > select First Aid in the top center and click Run.
Sometimes, the unsupported file system (NTFS file system)can bring your HDD not to mount on your Mac. You can format the HDD to Mac-compatible file format or you can use a third-party external drive mounter.
If you have no important files on the HDD, you can format the drive directly: open Disk Utility > click the icon of your unmountable external hard drive on the left sidebar > click on the Erase button function on the top of the Disk Utility window > repair this unmountable external hard drive with by reformatting > complete the related information and click Erase button to finish reformatting.
If you have files or data on the drive, iBoysoft NTFS for Mac driver is a better choice: free download, install and launch iBoysoft NTFS for Mac on your compute > connect your unmounted external hard disk to your Mac. Then, your external drive will be mounted automatically.
Hope this answer helps you out of trouble!