The problem, basically, is that while the force of the air on the propeller is the same as the force on the propeller of the air, the speed of the air differs from the speed of the propeller (both relative to the ground). So, somehow, the work done by the air on the propeller differs from the work done on the propeller by the air. (Probably, I'm not saying that right. Because I don't yet understand it properly.) And the difference is the engine power of the propeller, which we get "for free".
Ok, we need to distinguish between two different speeds, each of which might be called "the speed of the propeller". On the one hand, the propeller is attached to the cart, so "the speed of the propeller" could simply mean the speed of the cart. This is what I referred to above as "the speed of the propeller". On the other hand, the propeller is angled, and it's spinning, so that a fixed bit of air in contact with it slides along the propeller and gets pushed backward relative to the cart. The speed, relative to the ground, of this bit of air could also be called "the speed of the propeller", because, after all, it's in continual contact with the propeller, so the speed of the two must be the same.
What I called, in my quoted post, "the work done by the air on the propeller" used the latter, slower, 'speed of the propeller' which equals the speed of the air in contact with it, while what I called "the work done on the propeller by the air" used the former, faster, 'speed of the propeller' which equals the speed of the cart.
Let's call the slower speed "the speed of the propeller", and the faster speed "the speed of the cart".
Then, the work done by the air on the propeller (or on the propeller by the air; these are now identical) is the force of the air on the propeller, namely, 156 N, times the speed of the air at the propeller (or the speed of the propeller; these are now identical), namely, 5.9 m/s. The product is 920.4 W.
The work done on the cart is (156 N)(9 m/s) = 1404 J per second. However, this work should not be thought of as being done by the air on the cart, but rather by the propeller on the cart, because the air is moving slower than 9 m/s and acts on the cart only indirectly, via the propeller. The cart, on the other hand, and the part of the propeller in contact with the cart, namely, the hub, are moving at 9 m/s, and that interface is where the work on the cart is done.
But this is not the only work that the propeller does on the cart. That is just the work due to the forward force of the propeller on the cart, which tends to speed it up. The propeller, besides being forced forward by the air, also experiences a torque from the air tending to slow its rotation. This torque, transferred through the chain to the wheels rolling on the ground, tends to slow the cart. Therefore, this power (i.e., the product of the propeller's angular speed and the torque on the propeller from the air) is negative, and must be subtracted from the 1404 J per second to get the net work done on the cart per second by the propeller, which should equal the energy lost by the air per second, because the power transferred to the cart from the propeller comes ultimately from the air.
But we don't want to subtract too much. (That would make it seem as though the cart needed less energy from the air than it really does.) The actual torque on the propeller includes a portion due to rotational losses, to cover which does require energy from the air. So, we subtract the power that would be transferred through an ideal propeller which has no rotational losses. This is given by equation (4)
here, and equals, in our case, (156 N)(2 m/s + 1.1 m/s) = 483.6 W.
And, sure enough, 1404 W - 483.6 W does equal 920.4 W.
What's most confusing about this whole business, I think, is that, apparently,
some of the forces acting to slow the cart, namely rolling resistance and air resistance and friction in the chain,
increase the need for energy from the air, while
other forces acting to slow the cart, namely the backward force on the wheels due their spinning the propeller,
reduce the need for energy from the air.
I have the feeling that I'm comparing apples and oranges here, but that's the way it seems at first glance anyway.
I guess it depends on which direction we start from. If we start from zero, and add energy requirements, we need to add the energy required to overcome rolling resistance and air resistance and chain friction. And also the energy required to spin the propeller. But if we start with the (156 N)(9 m/s) = 1404 J of work done on the cart per second, we need to
subtract the energy required to spin the propeller. This must mean that somehow the 1404 J counts that energy twice!
Well, that was rambling enough. I will stop now and think about this yet some more.
I play to my audience. 
