I have a 2000 Ford Focus Wagon 4 cylinder and a.

It could be an exhaust leak near an oxygen sensor which will make the computer think the car is running lean. This will in trun make the car run rich and misfire. There also may be a reason for the broken exhaust. Your motor mounts may be broken. You can check them by putting the parking brake on, opening the hood to where you can see the engine from the drivers' seat and then put it in gear and give it a little gas. If it moves more than and inch you probably need motor mounts. You must replace them all at once along with the transmission mounts. I f you don't do this then the newest mounts will faill quickly.

P0171 is system too lean Bank 1
P0113 is Intake Air Temperature Circuit

So the system probably runs okay when you pull the pipe because it has that exhaust leak near oxygen sensor and thingks it is running lean when it is actually dumping in gas to compensate and is running rich. So, extra air leans it back out.
The intake air temperature sensor is bad and that could explain a lot too.

I am giving you a walk through diagnostic to help you. Just use the link below;

https://www.2carpros.com/articles/engine-misfires-or-runs-rough