If you have message hash (z) and signature (r,s) and you know that r.x is the same as your public key, then you can restore k and create any signature you want. There is no need to modify previous solution.
even that privatekey is the same as nonce
If you know some public key and you know r value of the signature, then you can always check is it the same or not, so you cannot remove that information from your challenge, unless you change it to some value where these two values are not related at all.