permanently deleting files from an external hard drive

Yes, you can do that. Please visit Pondini's Time Machine FAQ for help with Time Machine. This is an extensive site with most everything you may need for using Time Machine. You will not find a better resource for Time Machine in one location. Due to the passing of James Pondini, the site is no longer being updated. Therefore, some of the information may be a little outdated.

Drive Partition and Format - El Capitan or Later

Open Disk Utility in the Utilities' folder.

After Disk Utility loads select the external drive (out-dented entry with the mfgr.'s ID and size) from the side list.

Click on the Erase tab in the Disk Utility toolbar. A panel should drop down.

In the drop down panel set the partition scheme to GUID. Set the Format type to Mac OS Extended (Journaled.)

Click on the Apply button and click on the Done button when it is activated.

Quit Disk Utility.

Use the Terminal in the Utilities’ folder:

At the Terminal prompt enter the following:

rm -Rf [drag selected file(s) into the Terminal window]

Press RETURN.

Drag them to the Trash then Empty the Trash.

Are these Time Machine backups? If so they should be removed using the Time Machine application. That is the only proper way. Otherwise, erase the backup drive. This is the fastest and easiest way.