When "magic smoke" comes out of equipment, it usually means it's (burnt) toast. Replacing the TVS diode on the hard disk might work.
Presumably your "cord" is in fact, a mains power brick with a DC output and a voltage selector, e.g something like this? A cord is a passive device, i.e. a length of flex, wire, electric cable, etc.
No, this doesn't necessarily mean the drive is still receiving power. It just indicates the LED on the interface card is still working. The remainder of the circuit may be a charred mess.
By the sounds of it, you have an 8TB desktop drive with a USB 3.0 interface, as sold by Seagate or Western Digital. Inside the plastic case, you'll find a "standard" 3.5in SATA hard disk.
You cannot directly connect a USB cable to a SATA drive. Instead you need a special USB-to-SATA converter, which takes the form of a small printed circuit board, plugged into the back of the drive. This board converts USB to SATA and SATA to USB, so you can write data to the disk and read data back.
In the photo below, the USB-to-SATA converter is the small triangular shaped board in the top right hand corner. The larger rectangular(ish) green board underneath is part of the hard disk drive.
If you are VERY, VERY, lucky indeed, all you have destroyed is the USB-to-SATA converter board. The hard disk itself might still be intact.
To check if this is true, you could "shuck" the drive by breaking into the plastic case, pull out the hard disk, unplug the triangular USB-to-SATA board and make direct connections to the SATA data and power ports on the drive.
This video shows you how to "shuck" a drive. The terminology comes from the process used to remove oysters (live) from their shells.
Shucking is performed by people looking for cheap hard disks. Seagate and Western Digital do not make life easy. You are virtually guaranteed to break off some of the plastic tabs holding the two halves of the case together. This may invalidate the warranty, but you're past caring now.
Since the hard disk may also have suffered serious damage when "magic smoke" appeared, if you connect it directly to the motherboard inside a desktop computer, you might damage the computer. Try this at your own risk.
A safer option is to use a powered USB-to-SATA converter or docking station. Note: Unlike 2.5in USB drives which are "bus powered", a 3.5in drive needs +5V and +12V DC from an external source. The images below are for illustration only. They do not necessarily indicate your best option from local suppliers. If the adapter does not include +5V and +12V, your shucked drive will not power up.
After plugging the hard drive into the adapter, connect the USB cable to the computer and see what happens. If it works, give $20 to your favourite charity and be more careful next time (make backups).
One potentially serious "gotcha" is that in the past, some USB-to-SATA converters employed "sector translation". This meant you couldn't read the data off the drive without the USB converter card installed. Not common these days. The data was still there, but the computer couldn't read it, until you found a compatible adapter.
https://www.klennet.com/notes/2018-04-14-usb-and-sector-size.aspx
https://www.reddit.com/r/homelab/comments/hy83hm/sector_size_fiasco_and_recommendations_for/
I label the DC plugs on all my power bricks if they aren't +12V DC. I have some network switches than need +5V and a 10GbE MikroTik SFP+ optical switch with a +24V input. I hate to think what would happen if I plugged the 24V supply into the 5V switch. Magic smoke time.
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