That looks a lot like the water cut-off valve vacuum line going to the vacuum actuator. When the A/C system is in cold or MAX the valve gets vacuum through the HVAC control head to close the water valve to the heater.
Without it, hot water will always flow through the heater core, which will fight you by adding heat to the A/C box when you don't want it. Plus, it's a helluva vacuum leak.
Find the leak and fix it. You can have some success if you cut the broken ends parallel and then use some cold-weld electrical tape or similar to wrap around the "wound". This should buy you time until you can get a proper plastic tube replacement from the auto parts store or other plastic tube supplier. That's just one way to skin this cat. I've seen where some have used some rubber glue and shrink tube to seal the two pieces together instead of the tape, but it's a plastic tube so care must be given not to melt it and make it worse. You could use a small vacuum line and cut off what doesn't work and stick all that together as well.