  We investigate the Josephson current in a Fano-Josephson junction formed by
the direct coupling between two topological superconducting wires and their
indirect coupling via a quantum dot. It is found that when two Majorana zero
modes respectively appear in the wires, the Fano interference causes abundant
Josephson phase transition processes. What is notable is that in the presence
of appropriate direct and indirect inter-wire couplings, the fractional
Josephson effect disappears and then such a structure transforms into a
$0$-phase normal Josephson junction. On the other hand, if finite coupling
occurs between the Majorana bound states at the ends of each wire, the normal
Josepshon current is robustly in the $0$ phase, weakly dependent on the Fano
effect. We believe that the results in this work are helpful for describing the
Fano-modified Josephson effect.
