  We present a precise analysis of the Higgs mass corrections stemming from
vectorlike top partners in supersymmetric models. We reduce the theoretical
uncertainty compared to previous studies in the following aspects: (i)
including the one-loop threshold corrections to SM gauge and Yukawa couplings
due to the presence of the new states to obtain the $\bar{\text{DR}}$
parameters entering all loop calculations, (ii) including the full momentum
dependence at one-loop, and (iii) including all two-loop corrections but the
ones involving $g_1$ and $g_2$. We find that the additional threshold
corrections are very important and can give the largest effect on the Higgs
mass. However, we identify also parameter regions where the new two-loop
effects can be more important than the ones of the MSSM and change the Higgs
mass prediction by up to 10 GeV. This is for instance the case in the low
$\tan\beta$, small $M_A$ regime. We use these results to calculate the
electroweak fine-tuning of an UV complete variant of this model. For this
purpose, we add a complete $\textbf{10}$ and $\bar{\textbf{10}}$ representation
of $SU(5)$ to the MSSM particle content. We embed this model in minimal Gauge
Mediated Supersymmetry Breaking and calculate the electroweak fine-tuning with
respect to all important parameters. It turns out that the limit on the gluino
mass becomes more important for the fine-tuning than the Higgs mass
measurements which is easily to satisfy in this setup.
