Hello, Edwardsws!
Thanks for reaching out to the Apple Support Communities! That's a great question you have about allowing your child to download a specific application without allowing all applications. Based on what you're describing, we suggest checking out this article to Approve what kids buy with Ask to Buy.
With Ask to Buy, when kids want to buy or download a new item, they send a request to the family organizer. The family organizer can use their own device to approve or decline the request. For example, if a child wants to buy an app, the family approver can see the app and decide whether to allow it.
If the family organizer approves the request and completes the purchase, the item automatically downloads to the child's device. If the family organizer declines the request, no purchase or download will take place. If a child redownloads a purchase, downloads a shared purchase, installs an update, or uses a redemption code, the family organizer won’t receive a request. "
This will allow you to be notified when your child attempts to download an application and approve the app.
If this is not the path you're needing to go, feel free to circle back with more information about your issue.
Have a wonderful day!