  A high-accuracy time discretization is discussed to numerically solve the
nonlinear fractional diffusion equation forced by a space-time white noise. The
main purpose of this paper is to improve the temporal convergence rate by
modifying the semi-implicit Euler scheme. The solution of the equation is only
H\"older continuous in time, which is disadvantageous to improve the temporal
convergence rate. Firstly, the system is transformed into an equivalent form
having better regularity than the original one in time. But the regularity of
nonlinear term remains unchanged. Then, combining Lagrange mean value theorem
and independent increments of Brownian motion leads to a higher accuracy
discretization of nonlinear term which ensures the implementation of the
proposed time discretization scheme without loss of convergence rate. Our
scheme can improve the convergence rate from
${\min\{\frac{\gamma}{2\alpha},\frac{1}{2}\}}$ to
${\min\{\frac{\gamma}{\alpha},1\}}$ in the sense of mean-squared $L^2$-norm.
The theoretical error estimates are confirmed by extensive numerical
experiments.
