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In cluster-based quasicrystals, tetrahedra located in conventional Tsai clusters may be replaced by single rare-earth (R) ions at the cluster centers (pseudo-Tsai clusters). In this study, we investigate the effect of the pseudo-Tsai cluster incorporation on the magnetic structures of two approximants, the Tsai-type Tb-Au-Si [denoted TAS(0)] and Ho-Au-Si [denoted HAS(52)] with partial replacement of conventional Tsai clusters by pseudo-Tsai clusters, up to 52%. The mixture of Tsai and pseudo-Tsai clusters can be considered a different source of randomness/disorder other than the conventional chemical mix sites (Au/Si). The effect of the latter has been previously discussed regarding the origin/cause of spin-glass-like ordering and Anderson localization of electronic states in quasicrystals and approximant crystals. Single crystal neutron diffraction experiments at 2 K were performed and bulk physical properties (magnetization and specific heat) were investigated. In addition, earlier collected powder neutron diffraction data of TAS(14) with 14% replacement was reanalyzed in light of the results on TAS(0) and HAS(52). We find that the arrangement of ordered magnetic spins in the icosahedral shells of these phases is similar, while the cluster-center R magnetic states are different. In the case of TAS(14), the cluster-center Tb magnetic moments seem to affect the arrangement of surrounding icosahedral magnetic moments, and the magnetic structure of the icosahedral shell deviates from that of TAS(0). In the case of HAS(52), however, the icosahedral R magnetic moments are less affected by the cluster-center R, while the averaged cluster-center R magnetic moments are significantly diminished. We discuss these results considering the magnetic ordering effect on the bulk physical properties.