As to the first part of the current question: it was previously proposed that B receive $\operatorname{hash}(x)$ from A, then check $\operatorname{hash}(x+y)$ computed and sent periodically by C, comparing that against a recomputation made from $\operatorname{hash}(x)$ and $y$, using some property thought in the current question's title and second part.
That technique fails if C is dishonest. Problem is, C can compute $\operatorname{hash}(x)$ and discard $x$ (including while receiving and hashing $x$) and then use the very property used by B for verification. As a more easily solved aside, that technique also fails against an adversary having intercepted $\operatorname{hash}(x)$ sent to B, then impersonating C to B.
The problem is reminiscent of PDP: Provable (or proof of) Data Possession. Formalization of that problem is attributed to Giuseppe Ateniese, Randal Burns, Reza Curtmola, Joseph Herring, Lea Kissner, Zachary Peterson and Dawn Song with Provable data possession at untrusted stores (in proceedings of CCS 2007). A followup reference article by the same authors plus Osama Khan is Remote data checking using provable data possession, in ACM TISSec 2011.
Any PDP method can be used for the question's purpose by securely sending from A to B (with confidentiality and integrity) whatever the data originator of PDP is supposed to keep; but this does not match the requirement in this comment that B only holds public information.
As tho the original question (it's current second part): depending on the security properties thought for the hash, and the freedom we have about what $+$ is, what's asked is possible, or not.
For a hash with the security objective of being secure in the Random Oracle Model (informally: behaving like a random function), and all except very degenerate $+$, what's asked is clearly not possible, because it is a testable property that a random function would not have.
However, with common hashes like SHA-256, and $+$ taken as concatenation $\|$, something quite similar is possible: with $y$ starting with a certain prefix depending only on the length of $x$ (or $x$ ending with that), we can compute $\operatorname{SHA-256}(x\|y)$ from $\operatorname{SHA-256}(x)$ and $y$. That's the length-extension property (which breaks SHA-256 in the Random Oracle Model not adapted to allow the length-extension property, but is not against the originally stated security objectives of SHA-256).
Still with $+$ being concatenation, it seems possible to craft a hash without any restriction on the content of $x$ and $y$. However the solutions that I immediately see have some drawbacks, like a lesser security argument, constraints on the length of $x$, or/and reduced efficiency.