I saw one of the expansions of Euler Mascheroni constant in terms of Meissel Mertens constant as a consequence of Mertens theorem.
$$ B = \gamma + \sum_p \left\{ \log\left( 1 - \frac 1p\right) + \frac 1p\right\}$$
This is that expansion. Now I don't understand why is it difficult to prove the irrationality of Euler Mascheroni constant. Since we have infinitely many prime numbers, the sum over all those primes in the above equation, if converges must be a irrational, then why is it not considered as a proof of irrationality?
“if converges must be a irrational” is asserting what you need to prove. There’s no obvious reason for such a series to converge to an irrational number, and for all we know it may not.
Furthermore, even if this infinite sum is irrational, that doesn’t establish that $\gamma$ is irrational, since not much is known about the Meissel-Mertens constant you denoted by $B$.