We know this is a valid manipulation:
$$P(X, Y | Z) = \frac{P(Z | X, Y)P(X, Y)}{P(Z)}$$
and in general, if we have a constant "conditioned set" we can apply the standard manipulations for expanding joint distributions (for example) using the product rule of probability:
$$P(X, Y | Z) = P(X|Y, Z) P(Y|Z)$$
with the same logic applying for Bayes rule. Now, say we have the following:
$$P(X, Y | Z, H)$$
can we in general write:
$$P(X, Y | Z, H) = \frac{P(X, Z|Y, H)P(Y|H)}{P(Z, H)}$$
intuitively I imagine this would only be the case if we can guarantee certain independence properties?
EDIT:
Having worked out the algebra a little bit it seems that the only way we can say this is when $P(Z|Y,H) = 1$ ?