Determining structures of supported sub-nanometer catalytic clusters is important for finding binding sites, active sites, and reaction mechanisms. In realistic conditions, the cluster may be covered with adsorbed reactants and intermediates, which strongly affect the structure of the cluster compared to the low coverage regime. The global optimization of supported adsorbate-covered clusters is not possible with existing methods. We present a customized Basin Hopping (BH) approach, with a core (metal cluster)-shell (adsorbates) separation scheme, and per-form first applications to the search of low-energy isomers of Pt 7 /α-Al 2 O 3 at high hydrogen coverage (10H). The presence of adsorbates significantly alters the low-energy isomers compared to bare Pt 7 /α-Al 2 O 3 . Molecular dynamics simulations additionally show the different time-scales associated with the mobility of H and Pt atoms. The revised BH method with coreshell separation presents key advantages for the structural exploration of adsorbate-covered supported catalysts, relevant to realistic catalytic conditions.