Recently a functional limit theorem for sums of moving averages with random coefficients and i.i.d. heavy tailed innovations has been obtained under the assumption that all partial sums of the series of coefficients are a.s. bounded between zero and the sum of the series. The convergence takes place in the space $D[0,1]$ of càdlàg functions with the Skorohod $M_{2}$ topology. In this article we extend this result to the case when the innovations are weakly dependent in the sense of strong mixing and local dependence condition $D'$.