This letter focuses on integrating rate-splitting multiple-access (RSMA) with time-division-duplex Cell-free Massive MIMO (multiple-input multiple-output) for massive machine-type communications. Due to the large number of devices, their sporadic access behaviour and limited coherence interval, we assume a random access strategy with all active devices utilizing the same pilot for uplink channel estimation. This gives rise to a highly pilot-contaminated scenario, which inevitably deteriorates channel estimates. Motivated by the robustness of RSMA towards imperfect channel state information, we propose a novel RSMA-assisted downlink transmission framework for cell-free massive MIMO. On the basis of the downlink achievable spectral efficiency of the common and private streams, we devise a heuristic common precoder design and propose a novel max-min power control method for the proposed RSMA-assisted scheme. Numerical results show that RSMA effectively mitigates the effect of pilot contamination in the downlink and achieves a significant performance gain over a conventional cell-free massive MIMO network.