Place the barramundi pieces, curry paste, kaffir lime leaves, egg white, fish sauce and cornstarch into the bowl of a food processor. Pulse a few times to break everything up, scrape down the sides, then let it run until it forms a smooth puree, about one minute. Stir in the sliced Chinese long beans, then cover and place in the refrigerator while you prepare the cucumber relish.
Whisk together the sugar, fish sauce and vinegar in a medium bowl until the sugar has fully dis-solved. Add the cucumber, chiles, shallot and cilantro and toss to combine. Pour into a serving bowl and top with crushed peanuts if desired.
Rub your hands with a little bit of oil, then form the mixture into approximately 18 small patties. Heat 1/4 inch of vegetable oil in a large frying pan over medium-high heat. Fry the patties in batches until golden brown on both sides, then transfer to paper towels to drain.
Transfer the fish cakes to a platter with the bowl of cucumber relish on the side. Serve immediately.