Can you delete a game purchased from the Playstation store then re-install without a subscription at a later date?
If you delete a digital game that you have purchased through the PSN, you can redownload it from the store as long as you are logged into the account that purchased it initially. I constantly delete and redownload games depending on what I feel like playing, as I do not have enough storage space on my PS4 to hold all of the things I have purchased digitally, so I can confirm this myself.
As a side note, PS+ is never required to access content that you have purchased, however, content given for free as a bonus for being a Playstation+ member will expire if your subscription expires and you will lose access to it.
To see games that you own (downloaded or not), look at your library. It will have all of the software you have licenses for. See here:
Where to find your PlayStation Store purchases
Your most recently downloaded full games are added to the content launcher. If your purchase is not visible on the content launcher, you can find it in your [Library]. Your Library is arranged with all content grouped together, so if you have purchased DLC, go to the main title to find it.
Here is some confirmation of this:
Anything you've ever purchased is tied to your PSN account, so you can redownload any time you need.
There are several ways to re-download, but the easiest is to pull up your download history in the Store.
Here are the official Sony instructions on how to download digital copies of games.
How to download from the PS4 Library
- Select your local user account and go to [Library].
- Select the game or add-on you want to download (add-ons are in a ‘folder’ with the main title).
- Select ‘Download’ on the content screen.
As mentioned by Joe W, if the game has been removed from the store, you will not be able to re-download it once it has been deleted.
The only example I can think of like this is the Silent Hills demo P.T. that only a select number of users still have, since the demo was removed from the store and made unavailable for redownload.