  We study scenarios in the minimal and next-to minimal supersymmetric models
in which the lightest CP-even Higgs boson can have mass below the 114 GeV
standard model LEP limit by virtue of reduced ZZ coupling due to substantial
mixing among the Higgs bosons. We pay particular attention to the size of
corrections from superpartners needed for these scenarios to be viable and
point to boundary conditions at large scales which lead to these scenarios
while at the same time keeping electroweak fine tuning modest in size. We find
that naturalness of electroweak symmetry breaking in the mixed-Higgs scenarios
of both models points to the same region of soft supersymmetry breaking terms,
namely those leading to large mixing in the stop sector at the electroweak
scale, especially if we also require that the lightest CP-even Higgs explains
the Higgs-like LEP events at about 98 GeV.
