Most definitely. The hose you are referring to is most likely the drain designed to expel condensation from the A/C which is housed in the same box.
the heater core could be pluged or the heater control valve is stuck closed
could be a cracked radiator, bad radiator cap, a busted radiator or heater hose, bad heater core, or a busted fill box!
Its more than likely your heater core.
Check the thermostat or the radiator. The thermostat could be sticking or the radiator could be plugged.
its the heater core. but its not clogged. its cracked
You have to be more specific. What is not working?
Radiator fluid or antifreeze as it's called circulates through your heater core under the dash and provides you with heat and warm air for defrost. If you are smelling it inside the car then your heater core may have a leak. IS there any visible signs of leakage on the carpet in the car? The heater core usually has an inlet tube and an outlet tube through the firewall and maybe one of these is leaking. It could be leaking on the outside as well. Look for wetness around any hoses connecting to any tubes coming through the firewall. If you keep having to add antifreeze, you definitely have a leak.
Could be: Low coolant in the radiator - or - thermostat stuck open - or - heater core plugged.
Your car could be low on antifreeze or your heater core could be clogged. With my car, it was Heater matrix clogged with radiator sealant. The workshop cleaned it and it's start working fine!
Could be the heater. It's likely that most of the hoses to the rad were changed, but maybe a hose to a heater (if you have more than one), or the heater itself.
I wouldn't think you could flush a heater core. you can flush a radiator. Heater core you would have to replace if it is leaking or stopped up
heater hoses. or could be the run off water from your A/C unit