Let \((M, \eta)\) be an $m$-dimensional Nambu-Poisson manifold. We are interested in a Nambu-Poisson tensor $\eta$ whose order is smaller than $m$ and which has singularity at the origin. Then we compute Nambu-Poisson cohomology with respect to such a tensor $\eta$. More precisely, we compute Nambu-Poisson cohomology in case of \((\mathbb{R}^4,\eta)\) , where $\eta$ is a Nambu-Poisson tensor of order 3 with singularity at the origin. In particular, if $\eta$ is exact, we know that Nambu-Poisson cohomology coincides with relative cohomology, which has studied by B. Malgrange et al. [<6>]