The process of penetration of cobalt atoms through the h-BN nanomesh on Rh(111) is investigated with both spectroscopic and microscopic techniques. It is discovered that oxygen coadsorption can drastically modify the physical properties and behavior of the deposited Co clusters upon postannealing. In the absence of oxygen, Co forms small nanoparticles in the pores (bonding parts) of the h-BN nanomesh, which start to agglomerate at elevated temperatures without any considerable intercalation. However, even a tiny amount of coadsorbed oxygen reduces cobalt agglomeration and greatly promotes its intercalation and trapping under h-BN. The oxygen exposure necessary for a complete intercalation of 1-2 monolayers of Co is very low, and the formation of oxidic species can be easily avoided. The nanomesh structure remains intact upon intercalating submonolayer amounts of Co, while further intercalation gradually distorts and finally destroys the periodic corrugation. Fortunately, this process is not accompanied by damaging the h-BN sheet itself, and the original structure can be restored by removing Co upon annealing at higher temperatures.