 Solution

Substitution of $\delta x^a=ϵ{\psi }^a$ and $\delta {\psi }^a=-iϵ{\dot{x}}^a$ into the covariant component action leaves a total time derivative; the connection-dependent terms cancel by metric compatibility and the symmetry of the Levi-Civita connection. The boundary term vanishes for the stated decay. The Noether charge isup to the overall convention inherited from the supersymmetry parameter. Its graded square gives the Hamiltonian, $\{Q,Q\}=2H$ after canonical quantization.