We are interested in the following problem, of local nature. Consider a constant mean curvature (CMC) hypersurface M in an open ball B; assume that M is smooth and that taking the closure of M in B only adds a single point. In other words, we are looking at a CMC hypersurface with no boundary in B and with an isolated singularity. The question of interest for us is whether it is possible to approximate M by means of completely smooth CMC hypersurfaces, closed and without boundary in B, that have the same mean curvature as M. We look at the problem for non-zero constant mean curvature, since the case of the zero constant (minimal) was treated by R.Hardt and L.Simon. We want to identify hypotheses under which such an approximation is possible. It turns out that if M minimises the prescribed mean curvature functional on one side and under a mild assumption on the singularity (existence of a tangent cone that is smooth away from its vertex), such an approximation is possible in a suitably chosen small ball, in the Hausdorff distance sense and in fact in a stronger (graphical) sense away from the singular point. In order to prove the result under these assumptions we use techniques from both Geometric Measure theory and Elliptic PDE theory. We initially construct an approximation by solving a suitable minimisation problem, where each CMC hypersurface could have singularities in the open ball B. In order to study the possible singularities we develop a maximum principle for CMC hypersurfaces without boundary in B and with an isolated singularity of the same local structure as that of M. Once the maximum principle is established we use a blow-up argument, along the possible singularities, to prove that our approximation is in fact smooth. Finally, we show that there are examples of CMC hypersurfaces that satisfy all the assumptions needed for the smooth approximation property to be valid.