Before and after (for the 4th time). Repared this one yesterday with foam. I nail a piece of carboard to the back first. I then cut some pieces off old targets, put them inside the void, then fill with foam and nail cardboard to the other side. The cardboard will keep the foam from swelling and I think allows it stay more dense. Let it dry a couple days and it'll last a long time.
I really like the idea of the Nerf footballs. I'll try it the next time.
The key to making a patch last is to have a bunch of targets and spread the shots over all of them. I patched the coyote over 10 years ago and am still shooting it occasionally. I have 21 other patched targets but seldom set them up now.
I found a trick that I use with polyurethane glue, it expands like the foam when it interacts with water. While it's drying I push it back down into the repair, 3 or 4 times. It makes the material denser and and less of a mess to cut away outside of the animal later. It works pretty good.
I have 25 3 D targets and most have been repaired. For the larger holes I force old t-shirt in them. That helps slow the compound shot arrows down. I use the clear wrap that comes on a small cardboard tube and wrap around the target as tight as I can. I then poke a small hole in the wrap and use the Great Stuff foam to fill the gaps. Wearing rubber gloves I push the foam around to fill the gaps.
I also repair a lot of targets. One thin I do is to use a tube of silicon window type sealer, and spread it out like from a tube of toothpaste all over the target. I then use a scraper to spread it out into a thin layer all over the target. When it dries it is like have a new smooth skin on the target. You can get it in various colors or you can paint it. It makes the old faded wrinkled skin look almost new. Cost is about $5.