What to do if Mac mini fails to connect to router drive?

Have a look at > https://www.tp-link.com/us/support/faq/253/

FWIW I did the same with my NetGear Router for a while and it worked OK. Problem was that it was really slow, so I gave up and moved the hard drive back to one of my Mac mini's.

Based on what you have provided in your post, it does not appear that you have configured your TP-Link router properly to share out this attached hard drive.

As such, I suggest that you start by verifying your router's settings. Note: I do not have a TP-Link router, so I am going to provide you with some basic steps. This is where your router's user manual (or TP-Link online support) would come in handy.

1. Open your TP-Link router’s admin interface (usually `192.168.0.1` or `192.168.1.1`) in Safari.
2. Go to the USB Settings or Storage Sharing section.
3. Make sure the external drive is recognized by the router.
4. Enable SMB sharing. Take note of the network path or IP address the router assigns to the drive. It’ll often be something like: smb://192.168.0.1/VolumeName. NOTE: The `smb://Robet's iMac._smb._top.local` address sounds like it’s pointing to another Mac, not the router.
5. Go to Finder > Go > Connect to Server
6. Enter something like: smb://192.168.0.1 (Replace with your router’s actual IP if different.)
7. Hit Connect and see if it prompts for credentials or displays the drive.

Next, verify that your Mac's firewall and/or file sharing settings are not preventing your Mac from "seeing" this drive:

1. On your Mac mini: Go to System Settings > Network > Firewall – temporarily disable it for testing.
2. Also, ensure File Sharing isn’t interfering with the router SMB traffic.

If you’re still not seeing the drive, your TP-Link model might require setting up a user account for SMB access or only supports SMB1, which macOS dropped support for (by default). You can re-enable SMB1 using terminal tweaks, but that’s not ideal from a security standpoint. Better to check if the router supports SMB2 or SMB3 in its firmware.

Contact your ISP

@JM-Master

Curious ... how do you suppose that their ISP can assist them with this issue? The OP did not mention either the router or the attached drive as being provided by their ISP.