'Disk not properly ejected' on Time Machine drive

Sounds like power is being cut to the drive when the iMac goes into sleep mode.

As a test:

Turn on "Prevent automatic sleeping when display is off" and only allowing the display to turn off.

Then just leave the Mac on with display off and/or shut down for longer times of non-use.

+1 I agree with den.thed's advice. Going forward it is not wise to use "Walmart" quality EHDs as they have a high failure rate. When it's time, many experienced users on these forums use quality EHDs such as the https://eshop.macsales.com/shop/owc-mercury-elite-pro due to it's high quality, reasonable pricing and the level of support. These are not throw away enclosures like what you own now, which is good for the environment. When the HD dies on yours you throw the thing into a recycling bin and hope it's recycled responsibly. When the HD in a OWC drive dies you simply replace the HD (takes less than 5 minutes and a simple screwdriver) and you are back up in running in minutes. Then take the dead drive to your recycler.

I'd recommend that you buy the drive I recommended. This vendor specializes in Macs and everything they sell is extremely high quality, well tested and they offer the terrific support. I am a 20 year customer and have never had an issue with them. I am in no-way affiliated with them, just a satisfied customer.