We study eigenvalue distribution of the adjacency matrix $A^{(N,p,q)}$ of weighted random uniform $q$-hypergraphs $Γ= Γ_{N,p,q}$. We assume that the graphs have $N$ vertices and the average number of hyperedges attached to one vertex is $(q-1)!\cdot p$. To each edge of the graph $e_{ij}$ we assign a weight given by a random variable $a_{ij}$ with all moments finite. We consider the moments of normalized eigenvalue counting function $σ_{N,p,q}$ of $A^{(N,p,q)}$. Assuming all moments of $a$ finite, we obtain recurrent relations that determine the moments of the limiting measure $σ_{p,q} = \lim_{N\to\infty} σ_{N,p,q}$.