What should be the acceleration when a object is pushed by hand ?

I presume that you are confused whether to take acceleration due to gravity (g=9.8m/s^2) or some other acceleration.

Case 1: If the body on which you are applying force is moving along a horizontal direction, then the acceleration is calculated by the change in speed per unit time.
a= (v-u)/(t2 - t1)
v- final velocity of the body at time t2
u - initial velocity of the body at time t1

Case 2: If the body is moving in a vertical direction, then you will follow the same procedure and effective acceleration of the body is given by
g' = g+a, if body is moving toward the earth
g' = g-a, if body is moving away from the earth

Hmm....Getting your Point