I have a cedar deck/pergola that sits on a concrete slab. The deck and pergola are attached to the house with lags. A couple of the 4 x 4s suporting the deck (the joists are about 6 inches off the concrete)are situated on a section of concrete that has sunk about 4 inches toward the house's foundation. Short of tearing up the deck to tear out the sunken concrete, is there any other fix for this problem? Would a slab jacker be able to fix that section? Will the sunken crete cause any long term problems? My basement is very dry now and finished. Thank you.

You have mess on your hands. The sinking slab may eventually trap rain water and cause a damp basement. For the immediate issue, if you can get beneth the deck, just add additional 4x4 sistered to the existing 4x4s.

At some point your going to have to fix the concrete, it sounds like it may not of been poured deep enough in the corner.

I would temp brace the deck up and take out the concrete in this area and repour it, id go at least 8" deep where the weight is.
This really isn't a homeowner project here unless you have the concrete cutting tools at your disposal