Preheat the oven to 375 degrees F. Line a 12-portion 1/2-cup muffin tin with paper liners.
In a large bowl, sift the flour, baking powder, salt, cinnamon, nutmeg and ginger together.
In a standing mixer fitted with the paddle attachment, or with a hand-held electric mixer in a large bowl, cream the butter on medium speed until lightened. With the speed on low, slowly add the sugar. Increase speed to medium high and beat until light and fluffy, about 4 minutes. Add the eggs, one at a time, beating between each addition and scraping down the bowl as necessary with a rubber spatula. Add the carrot puree and orange zest and beat until well combined. With the mixer set on low, slowly add the dry ingredients and mix until just combined; do not overbeat. Remove the bowl from the mixer and with a rubber spatula, gently fold the cranberry relish into the batter until evenly distributed.
Evenly divide the batter among the lined muffin cups and sprinkle a little sugar over the top of each. Bake in the center of the oven until light golden brown and a toothpick inserted in the center comes out clean, 25-30 minutes.
Cool in the pan on a rack for 10 minutes. Remove muffins form the pan and transfer to a rack to cool completely. Serve warm or at room temperature.