Hello,

It appears to be liverwort which is indeed a symptom of wet and compacted soils. 

The best place to start is by simply pulling it out/prying it up with a hori hori knife or similar. You may be finding that the areas that you have covered with landscape cloth are not able to dry out. Given our very summer, it may be wise to pull that up, at least in the wettest areas, to try to let the ground dry out a little by letting the air get at it. The key is, though, not to walk on the wet ground and compact it further. If it's not possible to get to the liverwort because the ground around it is too wet, it won't hurt anything to let it be for a few weeks while (hopefully) the soil dries out a little. In fact, once the soil dries out, the liverwort may go away by itself.

Here are a couple of resources with further information: 

Liverwort | College of Agricultural Sciences (oregonstate.edu)

How can you control liverwort organically? - Cooperative Extension: Garden and Yard - University of Maine Cooperative Extension (umaine.edu)

I hope this helps. Thank you for reaching out to the UVM extension Master Gardener hotline!