  We consider a short Josephson junction with a phase discontinuity $\kappa$
created, e.g., by a pair of tiny current injectors, at some point $x_0$ along
the length of the junction. We derive the effective current-phase relation
(CPR) for the system as a whole, i.e., reduce it to an effective point-like
junction. From the effective CPR we obtain the ground state of the system and
predict the dependence of its critical current on $\kappa$. We show that in a
large range of $\kappa$ values the effective junction behaves as a $\varphi_0$
Josephson junction, i.e., has a unique ground state phase $\varphi_0$ within
each $2\pi$ interval. For $\kappa\approx\pi$ and $x_0$ near the middle of the
junction one obtains a $\varphi_0\pm\varphi$ junction, i.e., the Josephson
junction with degenerate ground state phase $\varphi_0\pm\varphi$ within each
$2\pi$ interval. Further, in view of possible escape experiments especially in
the quantum domain, we investigate the scaling of the energy barrier and
eigenfrequency close to the critical currents and predict the behavior of the
escape histogram width $\sigma(\kappa)$ in the regime of the macroscopic
quantum tunneling.
