Get some shade cloth it is relatively inexpensive or use some landscape fabric to cover part of the pen. Though I would suggest replacing the chicken wire with hardware cloth. The holes will let rats and mice in. Plus hardware cloth lasts longer.
Well, if you put a couple 2x4x8 or 10's across the top then you can set some plants on those boards and that will cast shade onto the space below, also when you water the plants the water will run into the enclosue which will offer some humid spots--which is good as well....also the shade cloth will offer some muted shade....