  We present a detailed study of the Higgs sector within an extension of the
Minimal Supersymmetric Standard Model that includes one Complex Higgs Triplet
(MSSM+1CHT). The model spectrum includes three singly charged Higgs bosons as
well as three CP-even (or scalar) and two CP-odd (or pseudoscalar) neutral
Higgs bosons. We present an approximated calculation of the one-loop radiative
corrections to the neutral CP-even Higgs masses ($m_{H_i^0}$) and the couplings
$H_i^0 Z^0 Z^0$ ($i=1$, 2, 3), which determine the magnitude of the
Higgs-strahlung processes $e^+ e^-\to Z^0 H^0_i$. Limits from LEP2 are then
considered, in order to obtain bounds on the neutral Higgs sector. Further, we
also include the experimental limits from LEP2 on $e^+e^-\to H^+H^-$ and those
on BR($t \to b H^+$) from Tevatron, to derive bounds on the mass of the two
lightest charged Higgs bosons ($H_1^{\pm}$ and $H_2^{\pm}$). Concerning the
latter, we find some cases, where $m_{H_1^{\pm}} \simeq 90$ GeV, that are not
excluded by any experimental bound, even for large values of $\tan\beta$, so
that they should be looked for at the Large Hadron Collider (LHC).
