This paper investigates the finite-thickness effect of two superimposed fluids on bubbles and spikes in Richtmyer-Meshkov instability (RMI) for arbitrary Atwood numbers by using the method of the small parameter expansion up to the second order. When the thickness of the two fluids tends to be infinity, our results can reproduce the classical results where RMI happens at the interface separating two semi-infinity-thickness fluids of different densities. It is found that the thickness has a large influence on the amplitude evolution of bubbles and spikes compared with those in classical RMI. Based on the thickness relationship of the two fluids, the thickness effect on bubbles and spikes for four cases is discussed. The thickness encourages (or reduces) the growth of bubbles or spikes, depending on not only Atwood number, but also the relationship of the thickness ratio of the heavy and light fluids, which is explicitly determined in this paper.