I have an Accord that I have to manually move the control valve under the hood to switch between using A/C and heat. The valve is fine and the cable moves, the problem is under the dash at the motor. I had a friend operate the temp **** from cold to hot and back while watching the motor operate. The motor also works properly but it isn't moving the linkage that pushes and pulls the cable. When I took the motor off the linkage moved freely and operated the control valve the way it should. Its very hard to say the least for a 6'2" man to work under the dash! lol I didn't see any broken parts but maybe mssing parts?? IDK Has anyone had this problem before? I plan to take it apart again after work tomorrow.

 

It may be due to a failing air mix control motor. The air mix control motor has several gears inside the unit, which is used to increase torque. Although the motor moves, it may not be generating enough torque.

 

Could be a failing internal plastic gear or one of the gears associated with operating the temp control valve. If you've got power @ the motor and the motor operates then it's likely a portion of the internal gearing. Also if the valve operates by hand fine then you know it's not a stuck or overly hard to operate valve and if you can move the cable by hand you know it's not a binding or stuck cable.
If it is an internal gear you might want to be 100% sure of the rest of the system though, the gears typically don't fail unless they are forced to operate something with an excessive amount of resistance.

I think this is the part that your talking about, it's called "Motor Assembly, Temperature Driver - Honda (79160-SDA-A01)" It's $83.79
I'm not sure if the gearbox is part of this motor part or not but it might be what you need.