This paper aimed to further introduce the finite element analysis of non-smooth data for semilinear stochastic subdiffusion problems driven by fractionally integrated additive noise. The mild solution of this stochastic model consisted of three different Mittag-Leffler functions. We analyzed the smoothness of the solution and utilized complex integration to approximate the error of the solution operator under non-smooth data. Consequently, optimal convergence estimates were obtained, and we also obtained the continuity conditions of the mild solution. Finally, the influence of the fractional parameters $\alpha$ and $\gamma$ on the convergence rates were accurately demonstrated through numerical examples.