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What year and model Ford are you working on?
Are both heater hoses hot going to the heater core at the firewall? once the engine heat up?
Try flushing the heater core. Since only one hose is hot, the heater core itself is clogged and not allowing hot coolant to flow through the heater core for heat. If it cannot be flushed out, then it will have to be replaced.
You will need to take the heater hoses off the heater core from under the hood and force water both directions through the heater core.
It is located behind the glove box but the heater hoses connect to it under the hood.
Yes, use a hoses and a nozzle to create pressure.
Yes, water should come out the other outlet. Then do it the other direction.