Know what you are doing.....You can write your name in a concrete drive with a power washer if you use too much pressured.

I hope you have better luck than I do with cleaning the underside of gutters. Nothing I can find removes that stuff.

You might want to repair the caulking before you power wash.

Another approach is to use one of the spray bottle products that attaches to your garden hose. That and some hand scrubbing should take care of the dirt.

I would power wash, being careful to not push water into the vented soffit. The rest should dry out, then recaulk. Be careful with the spray nozzle. You may need to pre-treat some areas.