Single Production of a Vector-Like Top as a Probe of Charged Higgs Bosons at a Muon-Proton Collider

We investigate the discovery prospects for a vector-like top partner ($T$) within the Type-II Two-Higgs-Doublet Model (2HDM-II) at future high-energy $\mu p$ colliders. The analysis focuses on the charged Higgs decay mode ${\mu }^{+}p\to {\nu }_{\mu }\bar{b}T\to {\nu }_{\mu }\bar{b}{H}^{+}b$, with the subsequent decay $H^{+}\to t\bar{b}$ yielding a final state with multiple $b$-jets and a charged lepton. A detailed detector-level simulation is carried out for benchmark configurations with charged Higgs masses around 600-650~GeV and vector-like top masses in the range $m_T\simeq 1.0$-$1.8$~TeV. For an integrated luminosity of 100~fb${}^{-1}$, discovery significances above $5\sigma$ are obtained across several benchmark points, remaining robust against systematic uncertainties up to 20%. At higher luminosities of 234~fb${}^{-1}$, the sensitivity exceeds $10\sigma$ for the lightest benchmarks and stays above $5\sigma$ even in the presence of 30% systematics.