If the dirt is level, you can cover it with anything. If it's 'bumpy' and/or uneven, then no matter what you cover it with, there will be 'unseen obsticals' (sp) that might get someone hurt if they don't see it.
As someone else said, you could be setting yourself up for a lawsuit. Even the best of friends with the best of intentioned friends will sue. To which you might say, "My homeowners covers me up to $500,000." Well, you better read the policy first to see if it covers negligence. In short, a good lawyer could argue that if they catch wind of your post here. Everyone has their ideas, including NOT having the kids on the dirt for safety reasons. If you 'know that', and don't take mitigating measures (like leveling it and posting signs and making sure the parents are WITH their kids and on and on and on), you'll lose.
In spite of it all, I'd keep them off the dirt, or better yet, have the party elsewhere. You can certainly reduce your 'exposure' by leveling the dirt and covering it w/hay and posting signs and telling parents and at your own risk and on.
My best to ya and hope this helps.
Jay J -Moderator
PS: God Bless America!
PPS: RE - the tarp: It will 'bind up' and cause someone to trip, and you're back to the lawsuit or a hospital visit, or both. Level the dirt and at least cover it w/straw and cover your legal tush as already stated, and then some ...