Deleted notes on iPhone

If you sync notes with iCloud (Settings > Your Name, iCloud > iCloud) and Notes are turned on once you deleted from the recently deleted folder they are gone. If you don't sync with iCloud and you have a backup from before you deleted the notes, you can restore your phone from that backup, but be advised, that will mean your phone will be current from the time the backup was made and new items on your phone (texts, notes, etc) will be gone.

Also, understand that once you delete something from the Recently Deleted folder, they are gone. That is why they temporarily go to the Recently deleted folder, in case you feel you made a mistake. It is like the Windows Recycle Bin. Once you empty that Recycle Bin, content is gone.

iCloud is not an archive. It is a syncing service.

Not once you empty the Recently Deleted. Do as lobsterghost1 said. If you have a backup made before you did this deletion, then restore that backup. No backup=No Notes.