You didn't mention the material used in your cover. Is it wood, vinyl, aluminum, synthetic? Each have different requirements.
The cover in picture looks identical to a product I have been selling for about 10 years. If it is same as picture you're in luck. This is known as a Pergola & is made of aluminum with a lifetime finish (powder coating). The factory finish is much more durable than any paint you could apply. Just spray on a good siding cleaner & power wash. Presto!
Looks like new again. Power washer with 3,000 psi or more has never been a problem.
If it is wood, do yourself a favor. Tear it down & start over. Painting is more trouble than it's worth. Besides, by the time it needs a second paint job, you will most likely have so much wood rot you will need to rebuild anyway.