Preheat oven to 350 degrees. Line a baking sheet with parchment paper or a silicone baking mat.
In a small bowl, combine the sugar, lemon zest, and candied ginger. By doing this, the sugar will coat the zest and ginger to keep them from clumping in the dough.
Combine the flour, baking powder, and salt in a large bowl, then add the sugar mixture. Give it a good mix. Set aside.
Add the lemon juice to the heavy cream and stir, then pour into the bowl with the dry ingredients. Mix until just combined.
Lightly flour a surface and knead the dough a few times until it comes together.
Pat out the dough into about a 9″ circle (about 1/2″ thick).
Cut into 8 wedges. Place each wedge onto the prepared baking sheet, leaving about one inch between them.
Bake in preheated oven for 20-25, until lightly golden on top. Leave on the sheet for 5 minutes after removing from the oven, then remove to a rack to cool completely.
To make the glaze: Mix the powdered sugar with heavy cream, 2-3 tablespoons of lemon juice, and 1 teaspoon of lemon zest. If the glaze is too thin, add a little more powdered sugar but if it’s too thick, add a little bit more cream. Drizzle glaze over cooled scones and top with more diced candied ginger.