  We investigate the effect of a time-reversal breaking impurity term on both
the equilibrium and non-equilibrium critical properties of entanglement entropy
(EE) in a three-spin interacting transverse Ising model which can be mapped to
a one-dimensional p-wave superconductor with next-nearest-neighbor hopping. Due
to the presence of next-nearest-neighbor hopping, a new topological phase with
two zero-energy Majorana modes at each end of an open chain appears in the
phase diagram. We show that the derivative of EE with respect to one of the
parameters of the Hamiltonian can detect the quantum phase transitions by
exhibiting cusp like structure at those points; impurity strength ($\la_d$) can
substantially modify the peak/dip height associated with the cusp. Importantly,
we find that the logarithmic scaling of the EE with block size remains
unaffected by the application of the impurity term, although, the coefficient
(i.e., central charge) varies logarithmically with the impurity strength for a
lower range of $\la_d$ and eventually saturates with an exponential damping
factor ($\sim \exp(-\la_d)$) for the phase boundaries shared with the phase
containing two Majorana edge modes. On the other hand, it receives a linear
correction in term of $\la_d$ for an another phase boundary. Finally, we focus
to study the effect of the impurity in the time evolution of the EE for the
critical quenching case where impurity term is applied only to the final
Hamiltonian. Interestingly, it has been shown that for all the phase boundaries
in contrary to the equilibrium case, the saturation value of the EE increases
logarithmically with the strength of impurity in a certain region of $\la_d$
and finally, for higher values of $\la_d$, it increases very slowly which is
dictated by an exponential damping factor.
