Heat a heavy saucepan and add oil, kumara and leek. Cover and fry over a moderate heat, stirring occasionally. Once kumara begins to colour, add chilli and ginger and continue to cook, covered, until kumara is tender.
Add water, thyme and salt and rapidly simmer, covered, for 10 minutes. Remove from heat and cool for 20 minutes. Purée soup to a fine texture, adding melon at the same time. Cool completely, then whisk in yoghurt and refrigerate overnight. The next day, season to taste.
To serve, ladle soup into chilled bowls, add a teaspoon of yoghurt and sprinkle some chives on top.