Security of hash functions over input extended with a random value Hashing a single message say $H(M)$ is vulnerable to a birthday attack.
But what if a combination is hashed, such has $H(M,R)$? Is it still vulnerable to the attack?

I think it is because $M, R$ can be assumed to be as $X$, so in essence $H(X)$ is being hashed and the birthday attack possibility still exists. Is this logic correct?
I think so cause $M, R$ can be assumed to be as $X$, so in essence $H(X)$ is being hashed and the birthday attack possibility still exists.


Yes, that's correct.

Note that there is no such function as $H(M, R)$, you'd need some way of combining the values; lets assume concatenation though (so $H(M | R)$).