Mix up your cleaning solution. If you have about four gallons of water in the bucket, then you will likely need less than a cup of flagstone cleaner. However, this is subject to the concentration of your cleaner, so read the package for specific measurements. Stir the cleaner into the five-gallon bucket and the water with the stiff push broom.
Apply the cleaning mixture to the flagstone. Use the stiff push broom to place the mixture on the flagstone. Make sure that it is in all the mortar and that it covers the stone. Put a little extra on any stained areas.
Let the mixture sit for about five minutes. This will give it time to work on the stains.
Scrub the flagstone patio. Work especially hard on stained areas. The stains should lift up fairly easily using the push broom like a scrub brush. If you need more cleaner, then simply redip the brush.
Rinse off the flagstone patio using the garden hose. Make sure that you get all of the cleaning fluid and water off the stones. This will reveal "problem areas" that may have stains that need to be revisited.
Use the push broom and some more cleaning mixture to remove any remaining stains. A second round of soaping and scrubbing will nearly always do the trick. Remember to rinse off the repeated spots once you are done.
Let the flagstone patio dry completely. This may take a few days. You may notice that there may be still some sand and even some small rocks on the patio after the scrubbing. This is okay as long as there are no more stains.
Sweep off the flagstone patio with the broom. Now your patio is spotless and ready for use.