Yes, cut the canes now. It will take 40-45 days, if you want to save the plant and have it bloom again. Don't wait. It's like cutting the deadends off your hair to make it healthy and grow again. Just do it, if nothing else.
The longer you wait, you risk letting the plant die.
Use the manure if it's compost, and work it into the soil.
But you also called it mulch. Mulch is for top of soil to keep moisture and weeds down. If it's mulch, you need
compost and sand for better drainage, and to lighten up that clay. And you need a good rose fertilizer. That rose is showing you illness as a result of it's living conditions.
It's like you've expected it to florish on bread and water.
It didn't. So now you have to give it medicine to make it healthy again.
Keep the grass clipping for the compost pile. Good green matter, mixed with brown matter will help everything in the spring. But it needs to breakdown first.
If you dress it now with grass, you'll truly find out what a horrible smell rotting grass can give off. And you'll never forget that smell! And isn't that rose by your front door? Not an appealing thought passing that to get in the house!
Again, good luck, and keep us posted.