You are asking the queue $Q\to (Q+Y,0)^+$ to be ergodic, where $Y$ is your $X-C$, and the stationary distribution of this queue to be integrable. Ergodicity requires that $E(Y)<0$, i.e. $\mu < p$. Integrability holds as soon as $Y$ (or, equivalently, $X$) is square integrable.
Amongst many other places, you might want to check example I.5.7 of Applied Probability and Queues by Søren Asmussen. (Are you sure this is not HW?)