If you put a covered patio next to the sunroom, it will block the sunlight from entering the room—no more sunroom. Orientation is irrelevant. Instead, build the covered patio away from the house.

What do you want from the new structure? A grilling and dining area? For sitting during the summer? A combination of activities? Each activity requires different amounts of space, and you should think about that before you get started.

How far away from the sunroom would be good distance to maintain the integrity of sunroom.i can’t put too away though

How far is too far? Does someone in your house have a disability? Or do you want to run electricity to the building? An electric line would be best run in underground conduit, ideally, as a separate ground fault circuit. It depends on what load you intend. Lights, ceiling fan and a few outlets for nothing bigger than a TV don't pull that much power. I don't know if there is a limit to the distance a line can be run. Talk to an electrician about this.

A pergola does let in more sunlight; how much depends on how far apart the slats are above the joists. Of course, a pergola lets rain through, so it's only usable when the weather is dry. An alternative is to use clear PVC or polycarbonate roof panels over the pergola slats. This material lasts from 10 to 20 years, becoming brittle with age. It doesn't provide shade by itself. Be sure the roof slopes to send the water away from your house.

Or put sky lights or a louvered ceiling in the covered porch to allow sunlight to come through.