  The kagome lattice is very attractive as it can host many novel quantum
states, such as the charge density wave, superconductivity, quantum spin
liquid, etc. Meanwhile, iridates often exhibit a strong spin-orbit coupling
(SOC) effect due to the large atomic mass of 5$d$ elements, which has important
implications for both the energy bands and the pairing symmetry of
superconductors. For the Laves phase superconductor Sc$_2$Ir$_4$ with a kagome
lattice, by doping Si to the Ir sites, we observed a nonmonotonic and two-dome
like doping dependence of the superconducting transition temperature $T_{\rm
c}$, which is typically found in many unconventional superconducting systems.
Interestingly, for some samples, especially Sc$_2$Ir$_{3.5}$Si$_{0.5}$ with the
optimal $T_{\rm c}$, after the suppression of superconductivity, the
normal-state resistivity exhibits a semiconducting behavior; meanwhile, the
specific heat coefficient shows an upturn which follows the relation
$C/T\propto{\rm ln}(T_0/T)$ at low temperatures. Around the optimal doping, the
resistance measurements exhibit strong superconducting fluctuations. And the
superconductivity related specific heat can be fitted by the model of a
$d$-wave gap after subtracting the normal-state background. These strongly
suggest unconventional superconductivity and correlation effect in the samples,
which is mainly induced by a flat band near the Fermi level when considering
the SOC, as supported by the first-principles calculations. Our results reveal
a new unconventional superconducting system Sc$_2$Ir$_{4-x}$Si$_x$ with strong
correlation effects induced by the flat band in the kagome system with strong
SOC.
