Small weakly-bound droplets determine a number of properties of ultracold Bose and Fermi gases. For example, Efimov trimers near the atom-atom-atom and atom-dimer thresholds lead to enhanced losses from bosonic clouds... Generalizations to four- and higher-body systems have also been considered. Moreover, Efimov trimers have been predicted to play a role in the Bose polaron with large boson-impurity scattering length. Motivated by these considerations, the present work provides a detailed theoretical analysis of weakly-bound $N$-body clusters consisting of $N-1$ identical bosons (denoted by "B") of mass $m$ that interact with a single distinguishable impurity particle (denoted by "X") of mass $M$. The system properties are analyzed as a function of the mass ratio $\kappa$ (values from $\kappa=1$ to $50$ are considered), where $\kappa$ is equal to $m/M$, and the two-body $s$-wave scattering length $a_{\text{BX}}$ between the bosons and the impurity. To reach the universal Efimov regime in which the size of the BBX trimer as well as those of larger clusters is much larger than the length scales of the underlying interaction model, three different approaches are considered: resonance states are determined in the absence of BB and BBX interactions, bound states are determined in the presence of repulsive three-body boson-boson-impurity interactions, and bound states are determined in the presence of repulsive two-body boson-boson interactions. The universal regime, in which the details of the underlying interaction model become irrelevant, is identified. (read more)