Tsai-type quasicrystals and approximants are distinguished by a cluster unit made up of four concentric polyhedral shells that surround a tetrahedron at the center. Here we show that for Tsai-type 1/1 approximants in the RE-Au-Si systems (RE = Gd, Tb, Ho) the central tetrahedron of the Tsai clusters can be systematically replaced by a single RE atom. The modified cluster is herein termed a "pseudo-Tsai cluster" and represents, in contrast to the conventional Tsai cluster, a structural motif without internal symmetry breaking. For each system, single-phase samples of both pseudo-Tsai and Tsai-type 1/1 approximants were independently prepared as millimeter-sized, faceted, single crystals using the self-flux synthesis method. The full replacement of tetrahedral moieties by RE atoms in the pseudo-Tsai 1/1 approximants was ascertained by a combination of single-crystal and powder diffraction studies, as well as energy dispersive X-ray spectroscopy (EDX) analyses with a scanning electron microscope (SEM). Differential scanning calorimetry (DSC) studies revealed distinctly higher decomposition temperatures, by 5-35 K, for the pseudo-Tsai phases. Furthermore, the magnetic properties of pseudo-Tsai phases are profoundly and consistently different from the Tsai counterparts. The onset temperatures of magnetic ordering (T-mag) are lowered in the pseudo-Tsai phases by similar to 30% from 24 to 17 K, 11.5 to 8 K, and 5 to 3.5 K in the Gd-Au-Si, Tb-Au-Si, and Ho-Au-Si systems, respectively. In addition, the Tb-Au-Si and Ho-Au-Si systems exhibit some qualitative changes in their magnetic ordering, indicating decisive changes in the magnetic state/structure by a moment-bearing atom at the cluster center.