A generalized super Camassa-Holm equation is propsoed, and claimedto be integrable in the sense of admitting of a Lax pair and bi-Hamiltonian structures. Performing Dirac reduction on its Hamiltonian structures, we obtain a bi-Hamiltonian structure of the super Camassa-Holm equation introduced by Geng, Xue and Wu [Stud. Appl. Math. 130 (2013) 1-16]. Through an appropriate reciprocal transformation, we connect the generalized super Camassa-Holm equation with a negative generalized super KdV equation, which is also equipped with a Lax pair and a bi-Hamiltonian structure.