Normally when you get a hole like that the tire is just about worn through, and the advice it to toss it. For something like a 10-day-old tire, though it might be worth an attempt at a repair.
I'd start by gluing on a piece of FedEx envelope (or similar) on the inside. I can't tell how big the cut is, so I'll guess 1/4" to 3/8". For something that size, look for some Shoe Goo and glue the cut from the outside.
My last recommendation would be to watch where you're going. That might mean getting off the shoulder, staying on the track swept clean by cars, or choosing another route with cleaner roads. It sounds like you ride through way too much debris if you're going through new tires every month or so.