How to remove/prevent standing water at campsite???

I would probably pump that.

I use something like this when needed at an apartment building:

https://www.lowes.com/pd/Superior-Pump-Utility-0-25-HP-Stainless-Steel-Submersible-Utility-Pump/1000737618

As the water absorbs, note the low point to place the pump. Get some hose such as pool backwash hose to carry the water where it will run off without issue.

If this is a permanent spot where you own and control then I would be more inclined to re-slope the area as needed for a permanent solution.

This is the pump I have:
https://www.waterpumpsdirect.com/FloTec-FP0S4100X-Water-Pump/p72416.html