No, the work done does not depend on how fast an object is raised up.
Since, W=mgh
For the constant value of m and g, the work depends on height upto which it is raised.
Power is the physical quantity which is defined as the rate of doing work i.e.
P=W /t
Where, W=work done
and, t=time taken.
Suppose an old man takes 10 sec to do a particular work whereas a young man takes only 5 s to do the same work. Here, the rate of doing work of the young man is more than that of the old man. Thus we can say that the power of young man is more than that of the old man.