  In this work, we provide the first strong convergence result of numerical
approximation of a general second order semilinear stochastic fractional order
evolution equation involving a Caputo derivative in time of order
$\alpha\in(\frac 34, 1)$ and driven by Gaussian and non-Gaussian noises
simultaneously more useful in concrete applications. The Gaussian noise
considered here is a Hilbert space valued Q-Wiener process and the non-Gaussian
noise is defined through compensated Poisson random measure associated to a
L\'evy process. The linear operator is not necessary self-adjoint. The
fractional stochastic partial differential equation is discretized in space by
the finite element method and in time by a variant of the exponential
integrator scheme. We investigate the mean square error estimate of our fully
discrete scheme and the result shows how the convergence orders depend on the
regularity of the initial data and the power of the fractional derivative.
