How to download a PS5 game from your phone

Before one really do anything about game downloads, please ensure ones PS5 console is connected to the PlayStation app.

You get there by clicking the settings icon on the home screen of the app.
Open the app on your iPhone, iPad, or Android device and navigate to the Settings menu.
Next, scroll down to “Console Management” and choose “Link Console to App.”
You’ll load a list of consoles with every PlayStation you’ve ever signed up to.
Just go to Options menu (by selecting the gear icon in the top-right corner of the PS5 home screen)
Scroll to the bottom to “System.” After this, just go “Power Saving” and choose “Rest Mode Available Features” from the submenu.
Toggle the “Stay Connected to the Internet” option, which will allow your PS5 to update automatically while in standby mode.
There’s also a second toggle called “Enable Turn on PS5 From Network” that lets you turn on your PS5 with the app.
When you’ve connected your console to the PlayStation 5 settings, allow access the PlayStation mobile app on your smartphone or tablet and access your game library.
Game Library, choose the game you want to download.

If you don’t see it in the list of recently played games, switch to the “Purchased” list, which will automatically sort your library alphabetically.
After you have picked a game, you will see “Download to Console” choice. Immediately just below, you will see which console is linked to the app.
Click the “Download to Console” button, and you’ll see the progress of the download in the queue.