How can you tell what apps you have already approved your child to download?

To add to Soosie's info:

Ask To Buy applies to purchases. Once the app is approved for purchase, they do not need approval to download or re-download it.

If you are concerned about your kid's use of apps on their device, look into parental controls:

Use parental controls on your child's iPhone, iPad, and iPod touch - Apple Support

Hi there Marc-G1,

Welcome to Apple Support Communities. We see you have some questions about the "Ask to Buy" with a family share setup and when you approved items for your child's account. We'll link the best article to keep handy for reference below:

Apple what kids buy with Ask to Buy

To specifically answer your question, the approved item will show in your child's purchase history. You can view an account purchase history with these steps:

See your purchase history for the App Store, iTunes Store, and more

1. Open the Settings app.
2. Tap your name, then tap Media & Purchases. You might be asked to sign in.
3. Tap Purchase History.
4. Your purchase history appears. If you want to see purchases that you made more than 90 days prior, tap Last 90 Days, then select a different date range

Have a great day!