  We report on the effect of magnetic impurities on the microscopic
superconducting (SC) properties of the kagome-lattice superconductor
La(Ru$_{1-x}$Fe$_{x}$)$_{3}$Si$_{2}$ using muon spin relaxation/rotation. A
strong suppression of the superconducting critical temperature $T_{\rm c}$, the
SC volume fraction, and the superfluid density was observed. We further find a
correlation between the superfluid density and $T_{\rm c}$ which is considered
a hallmark feature of unconventional superconductivity. Most remarkably,
measurements of the temperature-dependent magnetic penetration depth
${\lambda}$ reveal a change in the low-temperature behavior from exponential
saturation to a linear increase, which indicates that Fe doping introduces
nodes in the superconducting gap structure at concentrations as low as
$x=$~0.015. Our results point to a rare example of unconventional
superconductivity in the correlated kagome lattice and accessible tunability of
the superconducting gap structure, offering new insights into the microscopic
mechanisms involved in superconducting order.
