  We theoretically study the Josephson effect in a superconductor/normal
metal/superconductor junction based on Kekul\'e patterned graphene. For the
Kekul\'e-O patterned junctions, a Fermi momentum-splitting Andreev reflection
at the interface can be induced by the off-resonant circularly polarized light
applied in the normal region, which results in the possible $\pi$-state. In
contrast, for the Kekul\'e-Y patterned junctions, the Fermi momentum-splitting
Andreev reflection is strongly suppressed due to the valley-momentum locking
and the junction always exhibits the 0-state. The dependence of the critical
current on the junction length and the illumination parameter of the light
field is also presented in detail.
