Remove curse just lets you get rid of the cursed item
While remove curse will eliminate curses put onto an item, it does nothing about curses built into a cursed item. It just lets the user get rid of the item (where cursed items usually prevent that).
Aside from that, wish might be able to do it. It’s not a listed effect, though, so troubles abound there. I certainly wouldn’t be making any use of wish with a DM who actually included a cursed item.
Just, a side note: have you ever discussed these random cursed items with your players? I personally despise them; I don’t think they add anything interesting to the game, and tend to muck up the story and plot with pointless sidetracking. Cursed items as macguffins? Cool. That sword in the dragon’s horde that randomly prevents you from continuing on your quest until you trek back to town? Not interested.
Plenty of other players are, I realize; if yours are, there is no problem. But frankly I don’t really think “just to get on my player’s nerves” is a particularly good reason to do anything in D&D. Honestly, if it’s detracting from their play experience, it really just seems rude. After all, your players are committing some portion of their free time to play your game; wasting it indicates that you don’t value their time. So I recommend checking on that.