What is significance of recovering the chaining value of a hash compression function? What is significance of recovering the chaining value of a compression function, when the message and its output are known? in other words, if a compression function $CF$ takes a chaining value $h_{in}$ and a message $m$ as inputs and outputs $h_{out}$.

$h_{out}=CF(h_{in}, m)$

Is it a bad property if given $m$ and $h_{out}$ to recover $h_{in}$?
This should be considered as a weakness of the compression function, which is not a preimage resistant anymore. Sometimes, but not always, this may imply an attack on a hash function or a larger construction using this compression function.

For instance, consider the prefix MAC scheme, which is constructed out of a hash function $H$ as follows:
$$
MAC_K(M) = H(K||M).
$$
The prefix scheme is secure for some hash functions, for example the SHA-3 (Keccak) hash function.

However, if the compression function used by $H$ is not preimage resistant, then an adversary given $MAC_K(M)$ may recover the chaining value
$$
CV=H(K),
$$
and construct a valid MAC for any other $M'$ even though he does not know $K$.