Partial proof of work: an hash function that validates parts of input, insensitive to the rest
If you give me enough information to evaluate $h$ on inputs of my choice, then I can easily check whether $h$ is sensitive to the $i$th bit of input --- just evaluate it on two strings that differ only in their $i$th bit and see if the outputs are the same. So it is not possible to hide this property in the description of a public function.