I am failing to backup my MacBook on my external hard drive.

If your Time Machine backup drive is failing, then you need to buy a new one to replace it. Unfortunately, you have yet to mention the actual Time Machine error message, so I don't know where to go further. I provided the essential Apple document links on Time Machine in which you might have found an answer to your problem.

If your Time Machine backup drive has become corrupted in some way, then you can reformat the drive. You can restart your backup from scratch with Time Machine.

macOS User Guide

If you can't back up or restore your Mac using Time Machine

Back up your Mac with Time Machine

How to use Time Machine to back up or restore your Macs