Majorana edge modes are candidate elements of topological quantum computing. In this paper, we propose a Floquet engineering approach to generate multiple non-Hermitian Majorana zero and π modes at the edges of a one-dimensional topological superconductor. Focusing on a Kitaev chain with periodically kicked superconducting pairings and gain/losses in the chemical potential or nearest-neighbor hopping terms, we found rich non-Hermitian Floquet topological superconducting phases, which are originated from the interplay between drivings and non-Hermitian effects. Each of the phases is characterized by a pair of topological winding numbers, which can in principle take arbitrarily large integer values thanks to the applied driving fields. Under the open boundary condition, these winding numbers also predict the number of degenerate Majorana edge modes with quasienergies zero and π. Our findings thus expand the family of Floquet topological phases in non-Hermitian settings, with potential applications in realizing environmentally robust Floquet topological quantum computations.