Validating that a hash is over a specific value for two hashes with additional (known but different) input
This is not probably what you had in mind; but this is easy to achieve with a slight expansion.

Let us assume we have a private, public pair, with private key $P$, and public $K$. We basically consider $P$ to your $M$, and take $K$ to be part of the public knowledge of the verification of the hash.

Then $H(M+X)$ is defined to be $\textrm{Sign}(P,\textrm{Hash}(X))$ for any desired hash algorithm. Let $C_i=\textrm{Sign}(P, \textrm{Hash}(T_i))$. Given the public $K$, one can then verify the legality of each $C_i$ (and of course, for publicly known hash-function, one can also calculate the hash of $T_i$ directly).