Yes, there are an infinite number of solutions, though your teacher will want you to choose the most obvious one.
When the force does work on the mass, that work can be converted into two forms:
- the potential energy of the object
- the kinetic energy of the object
If you apply a force of $800g$ then once the object has been raised the 2.4m it will still be stationary because the force is only just enough to counteract gravity and there is none left of to accelerate the object. So the kinetic energy will be zero and the only change will be the change in potential energy. This is obviously how your teacher wants you to analyse the problem.
However, suppose the force is $800,000,000,000g$ (there's no special significance to this number other than it's big!). In this case the object will be raised 2.4m, but it will also be travelling at a high speed. The work done will be a lot higher than in the first case because it increases the kinetic energy of the object as well as its potential energy.
Strictly speaking the question should have stated that the object starts at rest and ends at rest 2.4m higher. This would fully specify the problem.