Broadband quantum memories, used as temporal multiplexers, are a key component in photonic quantum information processing, as they make repeat-until-success strategies scalable. We demonstrate a prototype system, operating on-demand, by interfacing a warm vapour, high time-bandwidth-product Raman memory with a travelling wave spontaneous parametric down-conversion source. We store single photons and observe a clear influence of the input photon statistics on the retrieved light, which we find currently to be limited by noise. We develop a theoretical model that identifies four-wave mixing as the sole important noise source and point towards practical solutions for noise-free operation.