as cron

Cron can be a fickle beast ... good luck!

It's probably a good idea to first remove any relative paths in your sunpaper.sh config file for good measure.

So change:
wallpaperPath="$HOME/sunpaper/images/The-Desert"
cachePath="$HOME/.cache"

to something like: wallpaperPath="/home/path/to/sunpaper/images/The-Desert"
cachePath="/home/path/to/.cache"

Here's a writeup about how to use Cron to call a script every minute:
https://linuxhint.com/run_cron_job_every_minute/

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If you use this method, I'd also recommend finding a way to call sunpaper.sh -c once each time your window manager starts, to avoid any issues with cache on restart.