If you had purchased OnSong previously on the App Store, you can reinstall the app again for free on devices that use the same Apple ID of the original purchase. You can learn about reinstalling in our How do I delete and reinstall OnSong article.
First and foremost, before you delete OnSong make sure that you backup your library. This will make are that you can restore your library in the event that the library is deleted along with the app.
The important thing here is that when you go to reinstall the app, make sure you go to the App Store app on your device and go view your Account. You can then go to the Purchases section and locate the version of OnSong that you had purchased. This will likely show OnSong Pro since it was purchased on the App Store. You can then download and install the app directly from this screen.
If you search and download OnSong from the App Store, you may have installed the new free version. The free version requires you to create an account and choose a membership tier, Premium or Essentials, in order to access the features that you had used in OnSong Pro. Make sure that you download OnSong Pro that you had purchased previously to access the features you had purchased.