In two recent papers, we used the hadro-quarkonium model to study the properties of hiddencharm and bottom tetraquarks and pentaquarks. Here, we extend the previous results and calculate the masses of heavy-quarkonium-kaon/hyperon systems. We also compute the spectrum of hiddencharm and bottom tetraquarks with strangeness in the compact tetraquark (diquark-antidiquark) model. If heavy-light exotic systems with non-null strangeness content were to be observed experimentally, it might be possible to distinguish among the large variety of available theoretical pictures for tetra-and pentaquark states and, possibly, rule out those which are not compatible with the data. * jacopo.j.ferretti@jyu.fi † santopinto@ge.infn.it FIG. 1: Schematic representation of heavy-light hadroquarkonium (right) and compact tetraquark (left) states. bsbn J P C N [(SD, SD)S, L]J E th [MeV] J P C N [(SD, SD)S, L]J E th [MeV] J P C N [(SD, SD)S, L]J E th [MeV]