Pergola with climbing plants. Add some misters for when you want to sit outside.

I recently saw this patio on Houzz with retractable fabric which might work for you. Unfortunately I didn't make note of the architect (I think they are in San Francisco) for more pics. Here is what I have..

Where do you see view of back yard? while sitting on deck? Do you want something year round or summer only?

ETA - how wide is the deck?

Here's the link to the patio renovation posted below. It's a spectacular reno. https://www.houzz.com/magazine/patio-details-a-relaxing-front-yard-retreat-in-los-angeles-stsetivw-vs~49412194

I do love sail shades - part depends on the angle of the sun when you want to be out there but since there is only one smaller seating area i would be tempted to get a lovely standing umbrella that i could shift around depending on the sun and call it a day (sturdy & stylish ones are expensive but last forever) when you are not using it you can tuck it against the wall so it does not obscure the view

We have two quality matching umbrellas on our deck and they make a surprising amount of shade. Way easier than a structure, not to mention cheaper and portable. In the spring and fall we don't want shade. We chose thatched tiki umbrellas because we like a tropical look, but go with your style. Make sure you get something sturdy, ideally supported by a patio table.

Sail shades are great because you can take them down in the winter. You will need somebody to install two or maybe three really strong tall posts for it.

You could also do an awning with a clear roof, and design it so you can string decorative fabric underneath for added shade in the summer. I don't think it will look heavy at all, I see awnings like this on narrow porches all the time.

The advantage to having something with a real roof is that it will protect your patio from rain. We love ours -- our rug and furniture never get wet, it's so easy to keep everything clean!

I meant to add: that second picture (Palram) offers different sizes and styles, and has a ton of add-ons, such as canopies you can attach underneath and easily slide open/closed.

I think a fundamental problem may be that it’s a narrow space which is essentially closed off on three sides. The brick walls absorb heat and there is little opportunity for airflow. I would build a pergola. A solid roof would only enclose the space and obstruct the airflow more, i would also install a properly sized exterior rated ceiling fan - two if appropriate, to help move air and cool you

Too much hardscaping. I would remove the deck and use shrubs, ground covers and limited amount of pavers. The vegetation will aid in transpirational cooling. Depending on where the sun is when you want to use the area, then add some umbrellas, shade sails. arbors....

Good point, fraker. And I might suggest a water feature.

Sail shades aren't a good idea if you live in area that has high winds. I live in the Chihuahuan Desert and every time we have high winds, a variety of items go flying away.

I still like umbrellas both for sun protection and I love the color punch .You could look at motorized awning maybe

I dislike umbrellas for a small deck because they always seem like they're in the way. Plus they shade such a small area. You have to move them as the sun moves, and close them when not in use (unless you want wind to knock them over when you're not there). I had one with an insanely heavy base -- it was designed to hold two cinder blocks -- and a stiff breeze would sometimes threaten to tip it over anyway. Worse -- even though it had wheels, it was difficult to move.

I second the sail shade idea. They're lighter looking and they do the job beautifully.

My boyfriend had this same problem and is putting in wooden structures with shade cloth over it. It keeps it from being too heavy plus you still get the rain . He hired a contractor to build the wooden structure and place the shade cloth on it. It looks really good and cools everything down.

I havexa small motorized awning over my patio that i can put out in spring/summer/ rain and pull in when its chillier temps. i love mine

You can solve this problem with a motorized shade structure. This one purchase solves the problem. Since it’s motorized, you can have the sun when you like.

I added a smaller shade sail and lights to my front patio this summer & love it!

Agree with the recommendations for retractable awnings. I can highly recommend SunAir awnings. We have their retractable awning and solar screens, all automated, and they are great quality. We live in place that can get high winds so anything sail-like is out of the question! They have manufacturing facilities in Arizona and Maryland, so check for a local dealer.

A pergola would be pretty, it wouldn't necessarily look heavy, that would depend on the materials and spacing you use… but if it's that sunny the coverage you desire may cause you to lose the natural light the windows provide inside the house. In that case I would consider a retractable awning.

Contemporary Backyard Retreat · More Info

Do you know what direction this side of your house faces? If it's west-facing, you may find that, at some point in the day, the sun will shine in right under a regular flat-topped pergola and still bake your deck. You could build a shed style structure, which would be higher at the house and then angle down towards the fence, but I think shade sails could be a better and much less expensive solution. If someone in your family is handy, a shade sail or several would be an easy DIY project. You'd basically attach to the house and run to poles or posts at your fenceline. I like the look of overlapping triangular sails, but you can also get rectangles if you prefer that look. They come in many colors, but I'd go with white for your application, since a darker color might be a bit oppressive.

Palm Springs Residence · More Info

The Castaway House · More Info

Retractable awning has my vote and with so many styles to choose, there will be various price points. Choose a color that you would like to accessorize with for a coordinated look. Bonus is you can sit out in the rain!

I have a retractable awning and love it!

If you are budget minded, I would do a retractable awning. The earlier suggestion of ceiling fans is a good idea for air circulation. You can’t have them with an awning but one or two standing fans would help if you are too enclosed.

Trees!

TREES! morestems in big pots or 'roof shaped/umrella'trees. Trees do not only give shade they also lower the temperature up to 6 degrees Celsius on the patio.

Ficus hedge between you and your neighbors and an umbrella.

I would try a sail sun shade. We use one every summer on our small deck. It goes up in the spring and comes down when it starts to get chilly. It looks like you might need to install two poles opposite your house. These come is different sizes and colors. Good luck! https://www.amazon.com/Canopy-Rectangle-Outdoor-Facility-Activities/dp/B074NHPRY7/ref=sr_1_7?keywords=sail+sun+shade&qid=1661350082&sprefix=sail+sun%2Caps%2C57&sr=8-7