Can any object have momentum even if its mechanical energy is zero? Explain.

(NO) No,an object cannot have momentum if its mechanical energy is zero.
Mechanical energy is the sum of kinetic energy $(KE)$ and potential energy $(PE)$.
If the mechanical energy is zero,then $KE + PE = 0$.
Since kinetic energy $(KE = \frac{1}{2}mv^2)$ cannot be negative,and potential energy is typically defined relative to a reference point,for the total sum to be zero,both $KE$ and $PE$ must be zero.
If $KE = 0$,then $\frac{1}{2}mv^2 = 0$,which implies that the velocity $(v)$ of the object is $0$.
Since momentum $(p)$ is defined as the product of mass and velocity $(p = mv)$,if $v = 0$,then the momentum $(p)$ must also be $0$.