Cleaning water staining on brick work
First of all you want to start slow on this with a little spot cleaning to learn what works in your particular situation. I've used power washers with detergents or bleach with success in these situations but you do have to be careful. 

Sometimes the cure is worse than the disease. Often when you clean brick or any surface you can end up with uneven coloration unless you do the entire wall. What I would suggest is start small with a light bleach solution (1 cup to 1 gallon water) and a brush (not wire) and try it on a small area that's not conspicuous. 

Getting it to match the surrounding brick is the tricky part. You want elbow grease but not too much - and feather out the edges to blend it. Let it dry to see if it did the job. If the bleach solution doesn't work you might have to go to a commercial product. Prosoco is one but there are others. If you can get the coloration close but still not what you want you might be able to even it out further with power washing. But once again for best results you may need to power wash the entire wall. 

Also if you do power wash you'll need to be careful around windows and cover and protect lights and that external speaker you have. Your brick looks in good condition but avoid any areas with the power wash where the mortar is loose or has gaps - maybe around windows. Good luck!