  We introduce a stochastic integral with respect to cylindrical L\'evy
processes with finite $p$-th weak moment for $p\in [1,2]$. The space of
integrands consists of $p$-summing operators between Banach spaces of
martingale type $p$. We apply the developed integration theory to establish the
existence of a solution for a stochastic evolution equation driven by a
cylindrical L\'evy process.
