Ps4 – Can you transfer or “return” an undownloaded game on PSN? Accidentally purchased

If by deleted, you mean have the account removed entirely from the PSN, then yes, I would think you could no longer play any downloaded game, or access any downloadable content.

However, if by deleted, you mean your PSN account is still intact, but you're replacing your hard drive so you have to add your account again to the new drive, I think it should be fine. The PSN should still have whatever games and content you've downloaded tied to your account on their servers. So, I would think that once you've logged in again using your new hard drive, you would be able to re-download all of your games.

First to check to see if you're actually using the same account on both consoles. The fact that you can't see your transaction history makes me suspect that you might have created a new account for your PlayStation 4 instead of reusing you existing account.

Otherwise in order for a game that you bought from the PlayStation Store for the PlayStation 3 to be playable on your PlayStation 4 two things need to be true. The first that there must be a PS4 version of the game. The PS4 isn't backwards compatible with the the PS3, so any game would need ported to the PS4 for you to play it on you new console. The second thing is that the particular "product" you bought needs to include both the PS3 and PS4 versions of the game. Most games aren't sold with both versions bundled together like this, so you need to buy separate versions for each console. Most developers will want to be paid for the work they had to do to port the game to the PS4, so this shouldn't be too surprising.

For example, Assassin's Creed IV Black Flag is available on the PlayStation Store in both a PlayStation 3 version and a PlayStation 4 version. These are separate products you need to buy separately. An example of one of the few games that you can pay one price and get the PS3 and PS4 versions, along with the PS Vita version too, is OlliOlli.

If you look at where it says "Playable On:" on the store page you can see the difference between the games:

There also the PlayStation Now service you may have heard of. This service lets you play games on your PS4 (and other Sony devices) by streaming them to you console. This works by running the games on Sony's servers (modified PS3s) and streaming the video output to your console. Unfortunately your purchase of games on the PlayStation Store doesn't entitle you to play these games on PlayStation Now. You have to pay a rental fee if you want to play them using this service on your PS4 or other device.