  Complete one-loop results are obtained for the class of processes
chi^0_i->chi^0_j h_a in the MSSM where all parameters entering this process
beyond lowest order are allowed to have arbitrary CP-violating phases. An
on-shell renormalisation scheme is worked out for the chargino-neutralino
sector that properly takes account of imaginary parts arising from complex
parameters and from absorptive parts of loop integrals. The genuine vertex
contributions to the neutralino decay amplitudes are combined with two-loop
propagator-type corrections for the outgoing Higgs boson. In this way the
currently most precise prediction for this class of processes is obtained. The
numerical impact of the genuine vertex corrections is studied for several
examples of CP-conserving and CP-violating scenarios. We find that significant
effects on the decay widths and branching ratios are possible even in the
CP-conserving MSSM. In the CP-violating CPX benchmark scenario the corrections
to the decay width are found to be particularly large, namely, of order 45% for
a Higgs mass of 40 GeV. This parameter region of the CPX scenario where a very
light Higgs boson is unexcluded by present data is analysed in detail. We find
that in this parameter region, which will be difficult to cover by standard
Higgs search channels at the LHC, the branching ratio for the decay
chi^0_2->chi^0_1 h_1 is large. This may offer good prospects to detect such a
light Higgs boson in cascade decays of supersymmetric particles.
