  We prove an algorithmic regularity lemma for $L_p$ regular matrices $(1 < p
\leq \infty),$ a class of sparse $\{0,1\}$ matrices which obey a natural
pseudorandomness condition. This extends a result of Coja-Oghlan, Cooper and
Frieze who treated the case of $L_{\infty}$ regular matrices. We also present
applications of this result for tensors and MAX-CSP instances.
