Simply wielding a shinki would allow the god to destroy an ayakashi. Essentially, they combine their power with their shinki's in order to Rend the ayakashi, which is a sort of specific spell used when the ayakashi is caught off-guard or is sufficiently weakened.

Ayakashi can only be defeated by a god who wields a shinki with Rend, natural powers can help to weaken the ayakashi, but cannot ultimately destroy it.