  Stochastic space-time fractional diffusion equations often appear in the
modeling of the heat propagation in non-homogeneous medium. In this paper, we
firstly investigate the Mittag--Leffler Euler integrator of a class of
stochastic space-time fractional diffusion equations, whose super-convergence
order is obtained by developing a helpful decomposition way for the
time-fractional integral. Here, the developed decomposition way is the key to
dealing with the singularity of the solution operator. Moreover, we study the
Freidlin--Wentzell type large deviation principles of the underlying equation
and its Mittag--Leffler Euler integrator based on the weak convergence
approach. In particular, we prove that the large deviation rate function of the
Mittag--Leffler Euler integrator $\Gamma$-converges to that of the underlying
equation.
