We compute the second differential osp(n|2)-relative cohomology of the Lie superalgebra К(n) of contact vector fields with coefficients in the superspace of weighted densities on the (1, n)-dimensional real superspace, n > 1, where osp(n|2) is the orthosymplectic Lie superalgebra. We explicitly give 2-cocycles spanning theses cohomology spaces. This work is the simplest generalization of a result by Basdouri [ On osp(1|2)-Relative Cohomology on S1|1. Communications in Algebra 42:4, 1698–1710 (2014)].