Mix a gallon of water and 1/4-cup of soap in the bucket. You can stir the mix around with your hand and the sponge. The mixture should be slightly sudsy.
Scrub the cushions with the sponge. You can scrub as hard as you like until you get the mold and mildew stains off. The detergent should kill the mold and mildew while it is breaking down the stains.
Dowse any remaining stains with lemon juice. You may still have some pale stains remaining on the cushions once you have finished cleaning them. These should be saturated with lemon juice.
Let the cushions dry in the sun. The lemon juice will lift the remainder of the stain out of the cushion using the sunlight as a catalyst.