I think this is how it works.
Hay day puts files on your device. So, you can delete the game but those files are still "local" to your device. If you then decide to download the game again, it picks up where you left off using those files. Game center allows you to save it "in the cloud" as well as your device. This makes it portable to other iOS devices. It can allow you to play the same farm on multiple devices.
Then there is icloud and iTunes. You are encouraged to back up what is on your device to your computer through iTunes or to icloud. Most users, since I manned Apple's technical support phone lines for a year, do not realize that their device auto loads the photos to the cloud so you can get them on other devices but deletes the cloud copy after a month. So, both will keep 2 or 3 backups which will include those files. If you get a new or different iOS device, these backups will make it like the device that was backed up. So, if you have an iPhone with applecare plus, if you smash your phone, you will get another one at the Apple store or by mail. Restoring the backup will get back most of your settings and information including the hayday files.
Lastly, if you do not restore a back up on a new device or, like I did a couple weeks ago, restore the device to factory settings and set it up as new, without access to those hay day files through a backup, connected through game center or Facebook, I do not think hay day would be able to get it back. But I am not 100% sure. Normally, if you quit playing for a time, I doubt you would be able to remember all the nitty gritty things the in-app support would ask you to get your farm back.