Time machine doesn't allow the external hard disk to make a back up copy

How to setup and use Time Machine: Back up your Mac with Time Machine - Apple Support

I suggest you start from scratch and do all steps required, don't skip any.

Other than that, you need to provide more detail for a more comprehensive answer: https://discussions.apple.com/docs/DOC-5931

What have you done since you last got a successful backup?

What exactly happens when you try to backup? I do you know it failed? Any error message?