If your driver sees plastic bags in your recycling cart, they may not collect your recycling. They will leave you an "Oops" tag on your cart explaining the problem.
After removing the plastic bags, you can put the recycling out on your next pickup day. If you have too much recycling to fit in your cart after missing a week of pickup, you can put out extra for free. Put out extra recycling in your own can or a cardboard box labeled "recycling" so the right driver gets it.