If $m$, $p_1$ and $p_2$ are odd 
we will have $p_{n+1} \le (p_n + p_{n-1} + m)/3$ unless $p_n + p_{n-1} + m$ is prime.  Heuristically one should expect prime values of $p_n + p_{n-1} + m$ to be very rare if the $p_n$ are large, so it should be unusual for $p_{n+1}$ to get very much larger than $\max(p_n, p_{n-1}, m)$.  For example, I tried $m=1$ and odd primes $p_1, p_2 < 1000$.  In each case, the sequence eventually became periodic; the largest prime that occurred was $55289$ (for $p_1 = 809$, $p_2 = 449$).  However, I would find it surprising if it were possible to prove $p_n$ is always bounded.