A tarp can be problematic because it will flap something FIERCE (giving it at best a short lifespan, and not infrequently causing it to fly away and take part of your run structure with it) and also catch rain and collapse and make a general mess. If you do it, make sure it is well supported and pitched so that a minimum of water collects.
For hawks, cheap trellis/deer/garden type netting works reasonably well. A few people have had hawks get tangled or get through, but by and large it seems to work well, requires relatively little support, and is cheap. It will come down in wet snow unless you make suitable arrangements, but it will probably not take your run fence WITH it when it comes down, unlike some other kinds of 'lids'.
Good luck, have fun,
Pat