Drain the cherries. Leave them in a strainer while you make the cookie dough.
Put flour, baking powder, soda and salt in a mixing bowl. Stir with a fork until thoroughly mixed. Cut in the softened butter with two forks, continuing to cut until the mixture looks like course corn meal (you can also do this in a food processor with the steel blade, using cold butter cut in 1/2 inch chunks).
Beat the eggs in a medium-sized bowl and combine them with the sugar.
Add the egg and sugar mixture to the rest of the ingredients and stir until thoroughly mixed.
Extract small bits of dough with your fingers and wrap them around each maraschino cherry, leaving the stem sticking out. Press the bottoms of the dough-wrapped cherries down slightly on a greased baking sheet.
Bake at 350 degrees for 10 minutes (cookies wil be white, if they start to brown, reduce the baking time). Let the cookies cool on the cookie sheet. Then dip them in powdered sugar so that the entire cookie part is covered, but not the stem.