Despite its unique structural features, the magnetism of single-layered cuprate with five oxygen coordination ($T$*-type structure) has not been investigated thus far. Here, we report the results of muon spin relaxation and magnetic susceptibility measurements to elucidate the magnetism of $T$*-type La$_{1-x/2}$Eu$_{1-x/2}$Sr$_x$CuO$_4$ (LESCO) via magnetic Fe- and non-magnetic Zn-substitution. We clarified the inducement of the spin-glass (SG)-like magnetically ordered state in La$_{1-x/2}$Eu$_{1-x/2}$Sr$_x$Cu$_y$Fe$_{1-y}$O$_4$ with $x = 0.24 + y$, and the non-magnetic state in La$_{1-x/2}$Eu$_{1-x/2}$Sr$_x$Cu$_y$Zn$_{1-y}$O$_4$ with $x$ = 0.24 after the suppression of superconductivity for $y$ $geq$ 0.025. The SG state lies below $sim$7 K in a wide Sr concentration range between 0.19 and 0.34 in 5$%$ Fe-substituted LESCO. The short-range SG state is consistent with that originating from the Ruderman-Kittel-Kasuya-Yosida interaction in a metallic state. Thus, the results provide the first evidence for Fermi liquid (FL) state in the pristine $T$*-type LESCO. Taking into account the results of an oxygen $K$-edge X-ray absorption spectroscopy measurement $[$J. Phys. Soc. Jpn. 89, 075002 (2020)$]$ reporting the actual hole concentrations in LESCO, our results demonstrate the existence of the FL state in a lower hole-concentration region, compared to that in $T$-type La$_{2-x}$Sr$_x$CuO$_4$. The emergence of the FL state in a lower hole-concentration region is possibly associated with a smaller charge transfer gap energy in the parent material with five oxygen coordination.
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