An accelerated random sequential adsorption process is studied as a model of chemisorption on a line with precursor layer diffusion. In this process if the position first selected for deposition is occupied then the particle diffuses and is absorbed on the first vacant position it visits. For k-mer deposition exact results are obtained for the gap distribution function. Physically measurable quantities such as the average island size and the probabilities of island nucleation, growth and coagulation are calculated as a function of coverage and the saturation coverage is calculated as a function of k. The continuum version of this model is also considered and potential applications of the models are discussed.