Where is the mistake in this reasoning? 2 identical objects are moving with different constant velocities. Then, in turn  same forces act on them for some period. Both times, the same amount of energy is used to produce those forces. As the forces acted, the objects covered different distances. Hence the works done by the forces are different. So the objects gained different amounts of energy. 
So we have a contradiction: The same amounts of energy were tranfered to the objects, but they gained different amounts of energy.
 A: If a force $\mathbf{F}$ acts on object $A$ and on object $B$ for a time $T$, and if the objects have different initial velocities parallel to the force, then the work done on object $A$ by $\mathbf{F}$ is not equal to the work done on object $B$ by $\mathbf{F}$.
If "amount of energy used to produce those forces" means the work done by $\mathbf{F}$, then the contradiction is due to stipulating that the work done by the force on each object is the same while also stipulating that the objects have different initial velocities (which implies the work done by the force on each object is different).


Independently from the objects, in some procedure, a force was
  produced using some energy . Then that procedure was repeated. As the
  result, the same force was produced, and the same energy was consumed.

If I understand your comment correctly, you're saying that there is some process that produces some linear mechanical power over some time period, and that this same process acts with the same force on two identical objects with different initial velocities.  But this where the contradiction lies.
Assume for simplicity that the process produces $P$ watts of power for $T$ seconds so that the energy 'consumed' is $E = P\cdot T$.
Now, the power associated with a force $\mathbf{F}$ acting on an object with velocity (in the direction of the force) $v$ is $P_F = Fv$.
Since the power associated with the force acting on each object is the same, the force acting on each object can only be the same if $v$ is the same for both objects.
A: You either have the same forces (i.e. same amount of Newton), then there are different energies that were transmitted. Or you have the same energy (i.e. same amount of Joule), then you will end up with different forces.
A force is the product of TWO objects, so you can't just say "the same force is acting upon both objects, independently of the objects". If the objects move in different velocities, and you "do the same thing" with them, then you end up with different forces.
A: The "amount of energy" needed to produce a given force for a given time depends on the velocity. Mathematically, this is just a fact, proven by the work-energy theorem, but I imagine you want an intuitive understanding.
Suppose that force is exerted by a rope, which wraps around a pulley with a weight on the other end. If the object moves faster, the weight falls further, though the force is the same in both cases. The energy released is proportional to the height loss of the weight, then the energy used does depend on the velocity. (If you don't believe that it takes more energy to lift a weight higher, than can you help me move into my new apartment on the 25th floor?)
A: I did some math and get an answer. Take 2kg masses, give v initial of 10 (100 J) and 20 m/s (400Joules) to each mass. Apply a loss of 50Joules to each mass over 10 meters, that equates to 5N force, therefore acceleration is -2.5m/s/s. And to get the distances to work out to 10 gives time of 1.18 and 0.518 seconds for each mass. So it works out.
So the question should have said it applies equal forces, over equal distances, for equal amounts of energy.  The time of application is different. So  greater power is required for the faster mass.
