How to Raise/Level a Sunken Patio Slab at the corner?
Since it's draining away from the house, I'd just correct the grade of the soil beyond it so that water isn't trapped and is free to keep flowing away; and leave it be. Dirt is easy to reshape - giant slabs of concrete, less so. If it's grass, peel back the sod, fix the level, roll the sod back down and water it well for a month.
There are ways it can be raised - &quot;mud jacking&quot; is the term of art, basically pumping grout under the slab to raise it hydrualically. Can be expensive, may be difficult to find a local who does it, may cost a lot to bring in a non-local to do it. So I'd pass. If the look of the pergola support is the hill you want to die on, consider that nobody but you will know if you make the pergola supports all the same height and let the top of the pergola not be level ;-). I bet nobody but you notices the difference in the posts if you do make it level...