How to lift/raise/repair a segment of concrete slab?
This slab is small enough that you should be able to easily lift it out and repair the base soil. Unless the concrete was weak initially or in otherwise poor condition it won't break. 


From the lawn side, trench along the slab about 6" wide and to the bottom of the concrete. Save the sod by wrapping it in a tarp. 
Using heavy steel or wood bars and some fulcrum blocks, begin lifting the concrete. Work in stages and block it up as you go. 
Once you're high enough, move the blocks rearward, underneath the slab. Continue maneuvering it until the entire slab is floating above the level of the surrounding slabs. 
Lever the raised slab sideways as needed to gain access to the soil below. It's not necessary to move it completely out of the hole area. 
Repair your soil base using non-organic sand or gravel. Tamp it well and leave it 1/2" higher than necessary. 
Move your slab back into position and remove the blocks. Be careful not to drop it to prevent damage to it and the surrounding slabs.


Allow the slab to settle for a few weeks, ideally through a rain event. If it doesn't rain, flood the area well to allow the soil to compact and settle. 

If you find that you've left it too high, a hose worked under the slab from the lawn side will allow you to wash out some fill soil. Work slowly so as to not go too far.