What is the best way to fill a 3 inch gap under my concrete front porch?
I'll assume that the hole is there because concrete was put onto unrammed gravel. The current state is left on the picture



Now assuming you've got rid of the animals you have to stabilize the existing concrete and the gravel beneath it. The easiest way would be to excavate some of the gravel until you reach some stable foundation (I'd guess it's around one foot deep) and put new concrete forming an extra "step". That will be the easiest way to force the new concrete under the old concrete.