I'm not quite sure why it's (Z-1) instead of just Z.
That page attempts to explain the Z-1. What part of it are you having trouble with?
wiki states it is "predominantly a consequence of the difference of the value of the electron-electron repulsion in the initial and final states of the atom that undergoes the x-ray transition"
but I'm not sure how we can derive the values of 1 and 7.6
so initially Moseley thought it was due to screening
Separate names with a comma.