Can you lose gas mileage if your catalytic.

There's a lot of variables, and given the lack of details or information, it is possible but in a round-about way. The fuel / air mixture is fine tuned by the Engine Computer based on the signals it gets from the front oxygen sensor. That sensor always comes before the catalytic converter. It measures the unburned oxygen in the exhaust system before the catalytic converter cleans it up. The second oxygen sensor comes after the catalytic converter to see how well the converter is doing its job. The signal from that sensor has no effect on adjusting the fuel / air mixture.

"Broken" can mean a lot of things, but if the exhaust pipe is physically cracked after the catalytic converter, the front O2 sensor won't pick that up. If the crack or break in the pipe is near or before the front O2 sensor, the momentum of the pulses of exhaust gas flow can introduce little pulses of vacuum that draw in outside air through that leak. The O2 sensor will see that unburned oxygen and the computer will interpret its signal as the mixture is too lean. In response it commands more fuel to all of the injectors. Even though more fuel goes into the engine, it can't burn completely because there's not enough oxygen in the air coming in. Now you have raw unburned fuel going out the tail pipe along with that air coming in through the leak. Oxygen sensors don't detect fuel; they only detect oxygen, so it is still going to report a lean condition while there's lots of wasted gas going out the tail pipe where you may smell it.

The introduction of air through a leak in the exhaust system is not a common occurrence but it is something we have to be aware of when looking for the cause of poor fuel mileage when all the obvious causes have been ruled out. It is much more common for a misfiring cylinder to cause the same problem. Unburned oxygen and fuel go into the exhaust system where only the oxygen is detected. Again, the computer responds by commanding more fuel from all of the cylinders. The symptoms can be a fault code set for "running lean too long", poor fuel mileage, you smell fuel at the tail pipe, and on '96 and newer models sold in the U.S. It should set a code indicating which cylinder is misfiring.

If by "broken" you simply mean the catalytic converter is not doing its job, that will not cause poor fuel mileage. That will be detected by the rear oxygen sensor and the fault code will be "catalytic converter efficiency".

Catalytic converters can become plugged too. That used to be caused by using leaded gasoline but today it is typically caused by a previous problem that caused too much raw fuel to go into the exhaust system. In an attempt to burn the fuel and clean up the exhaust, the catalyst can overheat and melt into a glob. You have to push harder on the gas pedal to keep the car going, if you even can keep it running, and that often results in engine overheating.