If you haven't seen it already, there's a great care sheet on this site by John
Frog Forum - Gray Tree Frog Care and Breeding
What is the injury?
More climbing spots and more plant cover would be my suggestion. A heat lamp isn't necessary, room temperature is fine unless your place is particularly cold. I use a plain old incandescent bulb at the top in the winter (my place
is cold in the winter!) with lots of perches at various heights so they can go where they're comfortable, but in the summer it's back to more energy efficient fluorescent bulbs.