Proving a Hash in Another Hash
I assume you mean H to be a one-way function. The way you 'prove' that you know half of the input of a one-way function (i.e. y or H(y) for the outer H()) will be to produce the other half of the input (i.e. x or H(x)) and show that H() of the two parts together gives you z.

In general, if your H is a Pseudo-random-function (which any good crypto-hash function is), then ANY possible combination of y and z has an infinite number of xs that will combine in the way you ask.