Can polynomial many-to-one reduction be done to a specific problem instance?
The mapping does not have to be surjective (onto) nor injective (one-to-one). In fact, any problem that can be solved in polynomial time can be polynomial time many-one reduced to any problem that has at least one accepting instance and at least one rejecting instance: just solve the original problem in polynomial time, then return the accepting instance if the original problem was an accepting instance, or the rejecting instance if the original problem was a rejecting instance.

That said, the Berman–Hartmanis conjecture states that all NP-complete problems are polynomial time isomorphic, meaning that there is a bijective polynomial-time many-one reduction between them with a polynomial-time inverse. This is currently an unproven conjecture, and only refers to NP-complete problems.