Plastic mesh? How did it get there? Is it something to keep the soil in place until the grass grows up? How big are the holes if you are worried about your
chickens' feet getting caught in it?
Since I've seen
chickens held in rabbit wire cages, chicken wire cages and various types of welded wire - without them getting their feet caught in it - I would put them out on it. You might also want to think about pulling it out of the sod, if at all possible, just my two cents here.