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We have deposited polycrystalline Re doped $(Fe_{65}Co_{35})_{100-x}Re_{x}$ (0 $\leq$ x $\leq$ 12.6 at\%) thin films grown under identical conditions and sandwiched between thin layers of Ru in order to study the phenomenon of spin pumping as a function of Re concentration. In-plane and out-of-plane ferromagnetic resonance spectroscopy results show an enhancement of the Gilbert damping with an increase in Re doping. We found evidence of an increase in the real part of effective spin mixing conductance [Re($g^{\uparrow\downarrow}_{eff}$)] with the increase in Re doping of 6.6 at\%, while a decrease is evident at higher Re doping. The increase in Re($g^{\uparrow\downarrow}_{eff}$) can be linked to the Re doping induced change of the interface electronic structure in the non-magnetic Ru layer and the effect interfacial spin-orbit coupling has on the effective spin-mixing conductance. The lowest and highest values of Re($g^{\uparrow\downarrow}_{eff}$) are found to be 9.883(02) $nm^{-2}$ and 19.697(02) $nm^{-2}$ for 0 at\% and 6.6 at\% Re doping, respectively. The saturation magnetization decreases with increasing Re doping, from 2.362(13) T for the undoped film to 1.740(03) T for 12.6 at\% Re doping. This study opens a new direction of tuning the spin-mixing conductance in magnetic heterostructures by doping of the ferromagnetic layerr, which is essential for the realization of energy efficient operation of spintronic devices.