  We give an elementary construction of a $p\geq 1$-singular Gelfand-Tsetlin
$\mathfrak{gl}_n(\mathbb C)$-module in terms of local distributions. This is a
generalization of the universal $1$-singular Gelfand-Tsetlin
$\mathfrak{gl}_n(\mathbb C)$-module obtained in [FGR1]. We expect that the
family of new Gelfand-Tsetlin modules that we obtained will lead to a
classification of all irreducible $p>1$-singular Gelfand-Tsetlin modules. So
far such a classification is known only for singularity $n=1$.
