If it is USB powered, it seems as if the USB you plug it into on your desktop is either faulty, sending surges through the hard drive. Is the desktop plugged into a surge protector? A laptop battery regulates the voltage going through, but you need external power surge for a desktop.
If the external drive is powered from the USB bus, and doesn't have a separate power connection, I would have to guess that your desktop computer's USB port does not supply enough power to run the drive, this is not unusual. If you have USB ports on the rear panel of your desktop system, you may give those a try, if you haven't already. Usually, the rear panel USB ports are "high power" ports, supplying the maximum current allowed under the USB specification. Often, ports on the front are connected to a USB hub internal to the computer, and will not supply the full spec USB power. I've seen this many times.
If your external drive has a separate power connection, then my previous explanation is not valid, and we would need more information to provide any useful answers.
It seems to be a Laptop HDD & I reply in that context.
3.5" HDD should have a separate Power Supply or Connector for it to be supplied from PC.
Details please
If powered from USB Port the it should have a cable Assy which has 1 connector on one side [should go to the External Drive] * 2 connectors on the other side.
Both should be plugged into 2 USB ports of PC /Laptop.
One is Power + Data & the 2nd only supplemented power .
If it works OK.
If not then the PCB in the Ext Box is @ fault.
Surely the you have to try another known to be a GOOD box.
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