  The aim of this work is to provide the first strong convergence result of
numerical approximation of a general time-fractional second order stochastic
partial differential equation involving a Caputo derivative in time of order
$\alpha\in(\frac 12; 1)$ and driven simultaneously by a multiplicative standard
Brownian motion and additive fBm with Hurst parameter $H\in(\frac 12, 1)$, more
realistic to model the random effects on transport of particles in medium with
thermal memory. We prove the existence and uniqueness results and perform the
spatial discretization using the finite element and the temporal discretization
using a fractional exponential integrator scheme. We provide the temporal and
spatial convergence proofs for our fully discrete scheme and the result shows
that the convergence orders depend on the regularity of the initial data, the
power of the fractional derivative, and the Hurst parameter $H$.
