  We present a study of the influence of disorder on the Mott metal-insulator
transition for the organic charge-transfer salt
$\kappa$-(BEDT-TTF)$_2$Cu[N(CN)$_2$]Cl. To this end, disorder was introduced
into the system in a controlled way by exposing the single crystals to x-ray
irradiation. The crystals were then fine-tuned across the Mott transition by
the application of continuously controllable He-gas pressure at low
temperatures. Measurements of the thermal expansion and resistance show that
the first-order character of the Mott transition prevails for low irradiation
doses achieved by irradiation times up to 100 h. For these crystals with a
moderate degree of disorder, we find a first-order transition line which ends
in a second-order critical endpoint, akin to the pristine crystals. Compared to
the latter, however, we observe a significant reduction of both, the critical
pressure $p_c$ and the critical temperature $T_c$. This result is consistent
with the theoretically-predicted formation of a soft Coulomb gap in the
presence of strong correlations and small disorder. Furthermore, we
demonstrate, similar to the observation for the pristine sample, that the Mott
transition after 50 h of irradiation is accompanied by sizable lattice effects,
the critical behavior of which can be well described by mean-field theory. Our
results demonstrate that the character of the Mott transition remains
essentially unchanged at a low disorder level. However, after an irradiation
time of 150 h, no clear signatures of a discontinuous metal-insulator
transition could be revealed anymore. These results suggest that, above a
certain disorder level, the metal-insulator transition becomes a smeared
first-order transition with some residual hysteresis.
