We study superconducting properties in multilayer thin films consisting of superconducting La$_{1.85}$Sr$_{0.15}$CuO$_4$ (LSCO) and Mott insulator Sr$_2$IrO$_4$ (SIO) and report enhanced superconductivity in optimized sample. These multilayer heterostructures show an increase in superconducting transition temperature ($T_C$) as compared to the single layer LSCO films. The temperature dependence of SIO single layer is also investigated under thermal activation, Arrhenius-type behaviour, and variable-range hopping mechanisms for different temperature regimes. The decrease in $T_C$ beyond an optimum thickness of LSCO in these multilayers is analyzed in the framework of a model based on the assumption of induced superconductivity in SIO-LSCO interface due to the doping of La and/or oxygen deficiencies into SIO layers
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