If a translation operator \hat T(\mathbf{x}) acts on the state |\psi\rangle, creating a new state |\psi_2\rangle then the expectation value of position for |\psi_2\rangle is equal to the expectation value of position for |\psi\rangle plus the vector x.
On the other hand, when the translation operator acts on a state, the expectation value of momentum is not changed.