Raising and extending a concrete shed base, does the existing concrete need to be broken up?
In your situation adding a new slab over the old has more chance for failure then success.  

Overlay slab might help underlying nonmoving cracks, but will not prevent active cracks in the existing slab from reflecting through.  

With a tree root causing the existing cracks the root growth will reflect into new slab if not fixed first.

Adding welded wire or rebar should be added to the overlay to help reduce cracks reflecting through.  

Better slope drainage is needed to keep moisture flowing away. 

One large issue is the expanded section of the new slab being partly on a different base material.  Compacted underlayment will hot have the same settling and expansion/contraction as the existing concrete slab.  This would become a prime area for breakage.  

The better approach is to break up the existing slab.  This material can usually be taken to a local sand and gravel pit for crushing and reuse as base material.   Once it is removed treat the tree root issue.
Lay base material, compact and pour a new slab.