Grease a loaf pan and line it with parchment paper, leaving some paper hanging from the sides to help pull out the loaf. Preheat the oven to 350 F.
Using a hand-held or stand mixer, cream the sugar and butter in a bowl, until light and fluffy.
Beat the eggs in, one at a time, making sure one is completely incorporated before adding the next one.
Stir in the milk, flour, baking powder, and salt until just combined. Do not overmix.
Fold in the shredded coconut and pour half of the batter into the prepared loaf pan.
Spoon raspberry jam on top of the batter, trying to spread it evenly over the top. Pour the remaining half of the batter evenly over the jam.
Bake uncovered, for 30 mins. Cover with foil and bake for an additional 25-30 mins or until a toothpick inserted in the middle comes out clean.
Remove from the oven and let cool for about 10 mins inside the pan. Using the hanging paper from the edges, carefully remove the cake from the pan and transfer to a cooling rack to cool completely.
In a small bowl make the icing by whisking the jam, lime juice, and powdered sugar together until smooth.
Pour the icing over the entire top of the cooled cake and sprinkle shredded coconut on top.