We show that the "Constant r-term Krichever-Novikov-type algebras" called by Nuyts and platten are derivation algebras of certain k-graded commutative algebra parametrized by Laurent polynomials, which implies that the algebras have natural central extensions. The affinizations associated with these commutative algebras are also introduced. More-Over, we give vertex operator representations of corresponding affine Kac-Moody algebras. By our vertex operator approach, we construct related k-graded Super-Virasoro algebras and their representations.