It is assumed that the eventual sink of global microplastic pollution is the deep sea. The primary vector for sediment and particulate pollutants to the deep sea are gravity currents down canyons along the coast line and at the shelf edge, and it has become recognised that these trap and transport microplastics. In order to quantify the potential storage within these marine environments, we develop a model of the transport of microplastic within turbidity currents. This model is to our knowledge the first to explicitly include microplastics and we find that the relatively simple model can produce turbidity currents similar to that observed within the Whittard Canyon; offshore Ireland. Based on this model we map the fate of microplastic within the canyon. Under most scenarios, the model simulates that small microplastics, fibres and fragments, will be transported into the canyon with little material leaving the canyon. Our best fitting model would suggest that at most only 11% of the source microplastic will bypass the canyon and be exported to the deep ocean floor. Marine canyons might therefore be a major sink of microplastic pollution, and act as a sponge between the anthropogenic source and the abyssal plane. This could have severe impacts on the ecosystems within these environments.