Peel the cucumbers and cut in half crossways. Divide each half lengthways, then use a teaspoon to scoop out the seeds. Chop the flesh roughly. Drop into a saucepan of lightly salted boiling water and cook gently for 2 minutes. Drain in a sieve, then refresh under cold running water.
Halve the melons and remove the seeds. Scoop out the flesh using a tablespoon and put into a blender or food processor. Add the cucumber, lemon juice and chopped mint, then whizz together until blended. Turn out the mixture into a bowl. Pour the water into the blender or processor and whizz briefly, then stir this liquid into the melon and cucumber mixture.
Stir in the prawns and season lightly with freshly ground black pepper. Chill for at least 1½ hours and up to 12 hours.
Just before serving, peel, stone and dice the avocado and add the diced flesh to the soup. Snip in some mint leaves to garnish and season to taste with more pepper if needed.