How do you remove an App from iCloud so your kids cannot download it without asking for permission?

To correct a misconception, Apps are not stored in your iCloud. The only place apps are stored, other than on a device, is the App Store. The cloud icon with the arrow refers to Apple's cloud, not your iCloud. Once an app has been purchased once it can be downloaded any number of times. If you have approved an app once I don't know of a way to unapprove it, although you might be able to set a restriction on it.

Or use an old-fashioned parenting approach. "If I find that app on your phone when I check it you lose the phone for a week."

Two things come to mind.

I don't know if this will work, but perhaps within Settings > Screen Time, you can set a restriction on the app. See especially "App Limits" here.

Use Screen Time to get insights about how you use your device - Apple Support

Another approach is that you may use Guided Access to restrict their activities, if this is appropriate for your situation.

Use Guided Access with iPhone, iPad, and iPod touch - Apple Support

I asked the same question earlier today! I wish it was easy to figure out. So frustrating. I've googled and have tried a few things but it is still there with the cloud symbol next to it.