The notion of $(m,n)$-ary hyperring was introduced by Davvaz at the 10$^{th}$ AHA congress [9], as the strong distributive structure. We generalize it, by introducing the notion of $(m,n)$-ary hyperring with inclusive distributivity. We present construction of $(m,n)$-ary hyperrings associated with binary relations on semigroup. We also state the condition under which there exists $(m,n)$-ary hyperring of multiendomorphisms for a starting $m$-ary hypergroup $(H,f)$. Finaly, we analyze connections between the obtained classes of $(m,n)$-ary hyperrings.