Preheat the oven to 425 °F. Line a baking sheet with parchment paper.
To make the glaze, combine the apricots in a saucepan with ¾ cup of the water, the maple syrup, chili powder, and cayenne. Bring to a boil and simmer for 10 minutes, or until most of the liquid has evaporated and the apricots are rehydrated.
Transfer the mixture to a blender and add the remaining ½ cup water. Blend until smooth.
Wrap each shrimp with prosciutto and place, seam side down, on the baking sheet. Brush with glaze.
Bake to 10 to 12 minutes, until the shrimp are opaque throughout. Brush again with the glaze midway through. Serve hot.