A weapon that has been destroyed through Enhancement or Enchantment can usually be restored with AP by visiting the Strange Traveler and clicking "Other", then "Destroyed Item".
Restored items can be Reforged by visiting Ferghus. This allows the item to be enhanced again, as long as the restored item is +10 - +14. An item must be Reforged before it is eligible to once again enter the Strange Traveler's shop.
If a weapon has been "Permanently Destroyed" this means that the weapon is unable to be revived by any means including with AP. If a weapon has been Restored before, and has not been Reforged, it will be Permanently Destroyed when the next Enchantment is failed.
In short, after the newest update, Enhancements and Enchantment can be failed as many times as you'd like, as long as you Reforge in between.
Keep in mind that a Restored item cannot be Enhanced until it is Reforged, but it can be Enchanted - be mindful of this when Enchanting your gear, as Enchantment failures will permanently destroy an item after it has been Restored!
You will know if a weapon or armor has been restored by noting the following tag in the tooltip:
Please keep in mind that the Strange Traveler only stores up to 10 destroyed items. If more than 10 are destroyed, the item that was destroyed first will be removed from the list.
Please know that this feature can only be accessed by characters that have reached level 90. If you have destroyed your weapon, but you are under level 90, you will have to wait until your character reaches level 90.
Note: Once restored, the item will be forever bound to the character.