= Solution
Substitution of $\delta x^a=\epsilon\psi^a$ and $\delta\psi^a=-i\epsilon\dot x^a$ 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> is
$$
\boxed{Q=g_{ab}(x)\psi^a\dot x^b},
$$
up to the overall convention inherited from the supersymmetry parameter. Its graded square gives the Hamiltonian, $\{Q,Q\}=2H$ after canonical quantization.
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