Mg2+ ions acts as cofactor for polymerase and it is also said that nucleotides which are added by this polymerase are also Mg2+ bound. Can k+ ions bind to the dNTPs or do they act as cofactor?
It is obvious that mg+ ion( divalent ion) is required for polymerase to be functional if not it will be apoenzyme and not functional at all. So there won't be any PCR amplification.
What is the mechanism that polymerase only requires divalent but not the monovalent cations to be functional ?
I can't fully answer your question, but it is definitely somehow connected with dNTP-Mg2+ interaction. It has to neutralise the negative charge on dNTPs and the DNA strand, and only one positive charge is not enough to do that.
I agree with previous comments that a PCR without a bivalent ion won't work. I have heard (but not tested it myself) that a PCR with manganese ions (Mn2+) will also work, but for instance with Ca2+ not. Too much k+(and other ions for that matter) will upset the ionconcentrations in the buffer reaction that comes with taq and in that way also inhibit the PCR.