Interesting example.
Yes, I believe that you could make the wing move up the pole. You would have to make sure that the wing has a positive angle of attack (AoA) since that is required to generate positive Lift. The AoA can be created by the shape of the wing or by tilting the wing back. And the engine would have to be able to push the cart fast enough to use that AoA to generate the Lift necessary overcome the Weight of the wing.
The standard equation for computing the force of Lift is: Lift (lb) = coefficient of Lift X dynamic pressure (lb/ft^2) X wing area (ft^2).
As illustrated above, a commonly used estimate is that the coefficient of Lift is 1/10 of the AoA. A commonly used estimate of maximum AoA is around 16 degrees - but that varies with each wing.
The dynamic pressure is 1/2 pv^2 where p is the air density (.00239 slugs at sea level) and v is the velocity (ft/sec). So the faster you go and the greater the angle of attack, the greater the Lift. Thus, for example, if the wing area is 6 ft^2, the cart is traveling at 30 mph (44 ft/sec), the air density is .00239 slugs and the coefficient of Lift is .25 (2.5 degree AoA), the wing would generate Lift of 3.45 lbs.
I should note that it is also possible to force the wing upwards even if you exceed maximum AoA - the same way you can move your hand upwards by sticking it out the window of a moving car and tilting it back. But that is not an example of true Lift, you are simply deflecting the wind downwards which shoves your hand upwards. The resulting Drag would be extremely high.
True Lift is the result of air pressure differentials. Within the limited AoA range, this develops enormous Lift and much less Drag. The Wright Brothers were able to fly because they understood the power of Lift and because they had created an engine with just enough power to generate the speed necessary to create enough Lift to exceed the Weight of their aircraft.
(EDIT - An interesting example of this difference can be seen with sailboats some of which are using wingsails to increase Lift.)
The title of your question might create some confusion because one could argue that you need "excess Power" to generate the speed required to create "excess Lift". In the literature, the term "excess Power" is used to describe the Power Available in excess of the Power Required for level flight at a constant speed. This excess Power can be used to either increase speed or to increase Lift at that speed.
However, I am limiting my response to the exact question you posed in the body of your discussion.