How to clean concrete?

There is no reason why bleach should make concrete disintegrate, but if you don't want to use it again you can use just about any all-purpose cleaner and a scrub brush. I have used a deck cleaner for boats, Pink Stuff, Dr Bronner's sal Suds, and plain old dish soap. They all worked fine, though the effects didn't last. You can get scrub brushes that screw onto a pole, so you don't have to get down on hands and knees to use a scrub brush. Even just wetting the concrete thoroughly and then scrubbing it with a brush and no cleaner makes a difference. However, you will probably end up doing it several times a year just to keep it looking passable. I have a concrete driveway apron that's much like that and there is no way to keep it looking decent unless I scrub every couple of weeks, which isn't going to happen. Even after I powerwashed it, it blackened again in weeks. Without changing the conditions it's in, which you can't do because the problem is that it's in shade, it will just keep blackening. Have you considered getting some cheapish fake lawn carpet to put there instead?

power washer

It's not the bleach which caused disintegration, it's the scrubbing action on poor concrete. Whatever you use, you need to be gentler.

muriatic acid but be informed before you use the stuff and follow all safety precautions on the container. here is a good example video. watch some of the others on the related list too.

https://www.youtube.com/watch?v=F4MQFaG-Vsw

There are products that you can spray on with a watering can, one of them is called "Wet and Forget". You should use it on a dry day, it will continue to improve for several weeks without further applications.

It's gonna be rough.

The Problem = That black stuff is mold, and the mold is retaining moisture.

As the seasons change, the North side of the house is not getting the same air circulation and evaporation
as the sunlit sides --- therein, as that moisture freezes and thaws, and expands and contracts with the seasons,
it weakens the porous surface structure of the cement. That's why you haven't been able to do anything
too drastic or "pressure-heavy" to the surface, including 'hard' brushing.

Bleach is recommended here ... then see what kind of surface damage is beneath it.
Go ahead and sweep it 'completely' or you're gonna have to pay for the same effort if someone else does it.

Take it from there *carefully* ... and, Good Luck!

try white vinegar , it prevents the spores from settling

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Hire somebody who knows how to do it right without killing himself.