  The comprehensive research of the electronic structure, thermodynamic and
electrical transport properties reveals the existence of inhomogeneous
superconductivity due to structural disorder in Ca$_3$Rh$_4$Sn$_{13}$ doped
with La (Ca$_{3-x}$La$_x$Rh$_4$Sn$_{13}$) or Ce
(Ca$_{3-x}$Ce$_x$Rh$_4$Sn$_{13}$) with superconducting critical temperatures
$T_c^{\star}$ higher than those ($T_c$) observed in the parent compounds. The
$T-x$ diagrams and the entropy $S(x)_T$ isotherms well document the relation
between degree of an atomic disorder and separation of the {\it
high-temperature} $T_c^{\star}$ and $T_c$-bulk phases. In these dirty
superconductors with the mean free path much smaller than the coherence length,
the Werthamer-Helfand-Hohenber theoretical model does not well fits the
$H_{c2}(T)$ data. We suggest that this can result from two-band
superconductivity or from the presence of strong inhomogeneity in these
systems. The multiband model very well describes the $H-T$ dependencies, but
the present results as well as our previous studies give arguments for the
scenario based on the presence of nanoscopic inhomogeneity of the
superconducting state. We also revisited the nature of structural phase
transition at $T^{\star}\sim 130-170$ K and documented that there might be
another precursor transition at higher temperatures. The impact of the magnetic
Ce-Ce correlations on the increase of $T_c$ in respect to the critical
temperatures of Ca$_{3-x}$La$_x$Rh$_4$Sn$_{13}$ is also discussed.
