I do not understand the significance of this 1566 Joules.
I don't think it makes sense to multiply the 174 N by the 9 m/s speed of the cart, because part of that 174 N is exerted by the ground on the bottom of the cart's wheels, which aren't moving at all relative to the ground, let alone at 9 m/s.
I'm not sure this objection is quite correct. It would seem to exclude from consideration the 125 W lost to friction in the chain (etc.) because that power also appears as a force at the bottom of the wheels, even though we do need to get this power from the wind. But, it appears elsewhere too, as we'll see.
When accounting for energy, as we are here, we should forget entirely about forces between the ground and the cart, because those forces do no work, and thus transfer no power. Let's analyze the 174.1-N force retarding the cart's motion into a number of parts, and see exactly where those parts do show up, and why it makes sense to multiply them by the 9-m/s speed of the cart when calculating how much power they extract from the cart.
(As always, forces come in pairs, due to Newton's third law of motion (action/reaction). We can view the 174.1-N force as a backward force exerted by other objects on the cart. Or we can view it as a forward force exerted on other objects by the cart. In what follows, I will take the latter point of view.)
The easiest force to think about is probably the air resistance. This is a force of 10 N, exerted by the cart directly on the air. The cart is moving at 9 m/s, so the power transferred is (10 N)(9 m/s) = 90 W. This power is dissipated as heat, and so needs to be gotten from the wind.
There is also the 53-N force of rolling resistance. This force originates in the flexing of the bottom portion of the tires and/or the deformation of the surface on which they're rolling, which are certainly not moving at 9 m/s. But as far as the body of the cart is concerned, it is a force which the cart body exerts on the axle of the wheels in order to cause it to move forward, at 9 m/s, thereby rolling the wheels along the ground. This force therefore converts another (53 N)(9 m/s) = 477 W of power into heat, and must also be gotten from the wind.
On to the 125 W lost to friction in the chain. Like rolling resistance, this friction makes the wheels harder to turn, because they are geared to the chain, and thus it also shows up as a force exerted by the cart on the axle. Again, the cart is moving at 9 m/s, so the force exerted is 125 W / (9 m/s) = 13.9 N. And, again, this power needs to be gotten from the wind.
The remaining 875 W, of the 1000 W that we supposed were required at the wheels to turn the propeller, corresponds in a similar manner to a force of 875 W / (9 m/s) = 97.2 N exerted by the cart on the axle.
We've accounted for the entire 174.1-N force, as 10 N + 53 N + 13.9 N + 97.2 N = 174.1 N. It would appear, however, that we've also accounted for 90 W + 477 W + 125 W + 875 W = 1567 W of power consumed, which is more than the 1380 W supplied by the wind.
However, and this is the important point, the power of 875 W, required to turn the propeller, is not dissipated as heat, but is transferred from the cart to the axle to the chain to the propeller
and then back to the cart: the propeller exerts a forward thrust of 250 N directly on the cart, at a speed of 2 + 3/2 = 3.5 m/s, for a power of (250 N)(3.5 m/s) = 875W transferred
to the cart. So this 875 W doesn't need to come from the wind, or anywhere else, because it never gets used up. It just travels around and around in a circle, from the cart back to itself.
Now, where did that speed of 3.5 m/s come from? It is the speed of the cart relative to the air in the immediate vicinity of the propeller. The cart moves at 9 m/s. The air moves 2 m/s slower, at 7 m/s. But the air near the propeller is slowed by it, by 3 m/s. This slowing happens partly in front of the propeller and partly behind it. Right at the propeller, half of the slowdown has occurred, and so the speed of the air there is 5.5 m/s, and the difference between this and the 9 m/s speed of the cart is 3.5 m/s.
Why is this speed the right one to use?
The propeller exerts a forward force of 250 N directly on the cart, and they're both travelling at 9 m/s. So the total power transferred from the propeller to the cart, via this forward force, is (250 N)(9 m/s) = 2250 W. Where does the propeller get this power from, to transfer it to the cart?
Part of it comes from the air. At the propeller, the air is moving at 5.5 m/s, and it exerts a force of 250 N on the propeller, so the power transferred is (250 N)(5.5 m/s) = 1375 W.
And the remainder of the power transferred to the cart by the propeller, namely (250 N)(9 m/s - 5.5 m/s) = (250 N)(3.5 m/s) = 875 W comes from the cart itself, via the chain from the axle, on which the cart pushes, as mentioned before.