The convergence of stochastic integrals driven by a sequence of Wiener processes $W_n\to W$ (with convergence in $C_t$) is crucial in the analysis of stochastic partial differential equations (SPDEs). The convergence we focus on in this paper is of the form $\int_0^T V_n\, {\rm d} W_n \to \int_0^T V\,{\rm d} W$, where $V_n$ takes values in $L^p([0,T];X)$ for some finite $p\ge 2$ and a Banach space $X$. Standard methods do not directly apply when $V_n$ only converges weakly in the temporal variable to $V$. We provide (weak) convergence results that address the need to take limits of stochastic integrals when only weak temporal convergence is available. This is particularly relevant for SPDEs with singular behaviour.