My phone was just re-charging and it's offline from the net.
The phone chirped to say a message had arrived.
I check the screen and find that DJIGo4 has sent me a message announcing that it will automatically stitch together videos once a flying session has been completed.

Really UNCOOL.

I do not want apps burning chip cycles announcing features. Especially when the app was closed!
I don't mind if the app wants to highlight a feature when I open it, but once I close it, I want it shutdown.

DJIGo4 seems to think it's a messaging app.
I got rid of Yahoo email app because it was running all the time. I don't think it was very well written cause it was slowing my phone operations.
My phone has been running a little slow recently and I'm beginning to think it due to DJIGo4 burning chip cycles by running in the background.

Just had a look in the Android settings and it tells me that DJIGo4 is always running. Why???
I've just 'forse stopped' it and I'm curious to see if it pops up again.

I believe what you got is a notification from the app instead of a message. You can go into your device setting and disable notification. Also, regarding the cpu cycle, most mobile OS just put the app on idle when we close them using home or back button so the only way to prevent the app from using cpu is to force close it.