There's no one correct/strict method per se to do it but multiple ways of cleaning engine bays. As long as you take precautions in covering up sensitive components and areas, it should be fine.
One area not specifically mentioned in this thread that shouldn't be overlooked is the intake at the front of the vehicle just slightly off to the right. I usually place a thick microfiber, towel in there to prevent any liquid from getting in. If you have an exposed K&N-style air filter, that should be covered with something.
A several weeks ago I sprayed diluted OPC APC and used a few old/worn/stained microfiber towels for wiping clean, a flag-tipped brush (which I use for the fender wells) and a small detail brush for agitation. I also have a daytona-style wheel brush and even a mini version which would do also the job. I clean my brushes after every session that they're not an issue of cross contamination - especially for dirty areas like engine bays.
I also used a spray bottle filled with distilled water for rinsing before drying.
After cleaning, you can apply a dressing such as Aerospace protectant or Black on Black (aerosol) - depending on the surface. It's not necessary, but I sometimes apply a durable sealant on the painted surface of the engine cover such as CG's Jetseal or Wolfgang 3.0.
Note that during that time I did not use a garden hose or pressure washer. You can if you want, but it's not the only way to do it - especially if all that is needed is some light cleaning.
Edited to add:
For earlier CX-5 models, an aerosol-type trim dressing such as Meguiar's Ultimate Black, Meguiar's Trim Detailer or CG's Black on Black make it far easier to dress the honeycomb grille.