  In this work we introduce a theory of stochastic integration for
operator-valued integrands with respect to some classes of cylindrical
martingale-valued measures in Hilbert spaces. The integral is constructed via
the radonification of cylindrical martingales by a Hilbert-Schmidt operator
theorem and unifies several other theories of stochastic integration in Hilbert
spaces. In particular, our theory covers the theory of stochastic integration
with respect to a Hilbert space valued L\'{e}vy process (which is not required
to satisfy any moment condition), with respect to a cylindrical L\'{e}vy
processes with (weak) second moments and with respect to a L\'{e}vy-valued
random martingale measures with finite second moment. As an application of our
theory of integration we prove existence and uniqueness of solutions for
stochastic stochastic partial differential equations driven by multiplicative
cylindrical martingale-valued measure noise with rather general coefficients.
