Year would be helpful - looks like an 00 or 01.

The hose appears to be a vacuum hose for an emissions component. You should be able to go to the hardware store or parts store and grab a length of vacuum hose with the same I.D. and just replace it.

Ok, I was wondering why it looked weird.. I was thinking I was looking at the area near the oil filter and not the firewall.. The cut rubber hose behind it was messing with me until I realized it was a heater core hose (as you've obviously bypassed the core).

I'm not sure of the exact part, but it definitely appears to be a vacuum hose (unless maybe a washer fluid hose).. regardless, you can replace the whole hose with some generic vacuum line from the hardware/parts store and be good.

That thing Is the evap purge solenoid.. It pulls the gas fumes from gas tank/charcoal canister.
Feeds the gas fumes into the intake manifold for the engine to burn the fumes off in combustion,, instead of allowing the fumes to escape into the air.. Save the planet crap..

little peace of rubber hose to attache the two ends together and call it a day.. Or just tape it extra good like you did.. As long as no air leaks happen at the break you will be good.

Take the fitting off and bring it into an auto parts store and tell them you need vacuum hose that fits over the tube. Cut a small section to bridge the gap. Zip tie both sides for better effectivity.