  It has been pointed out that supersymmetric extensions of the Standard Model
can induce significant changes to the theoretical prediction of the ratio
$\Gamma(K\rightarrow e\nu)/\Gamma (K\rightarrow\mu\nu)\equiv R_{K}$, through
lepton flavour violating couplings. In this work we carry out a full
computation of all one-loop corrections to the relevant $\nu\ell H^{+}$ vertex,
and discuss the new contributions to $R_{K}$ arising in the context of
different constrained (minimal supergravity inspired) models which succeed in
accounting for neutrino data, further considering the possibility of
accommodating a near future observation of a $\mu\to e\gamma$ transition. We
also re-evaluate the prospects for $R_{K}$ in the framework of unconstrained
supersymmetric models. In all cases, we address the question of whether it is
possible to saturate the current experimental sensitivity on $R_{K}$ while in
agreement with the recent limits on $B$-meson decay observables (in particular
BR($B_{s}\to\mu^+\mu^-$) and R($B_{u}\to\tau\nu$)), as well as BR($\tau\to e
\gamma$) and available collider constraints. Our findings reveal that in view
of the recent bounds, and even when enhanced by effective sources of flavour
violation in the right-handed $\tilde{e}-\tilde{\tau}$ sector, constrained
supersymmetric (seesaw) models typically provide excessively small
contributions to $R_{K}$. Larger contributions can be found in more general
settings, where the charged Higgs mass can be effectively lowered, and even
further enhanced in the unconstrained MSSM. However, our analysis clearly shows
that even in this last case SUSY contributions to $R_{K}$ are still unable to
saturate the current experimental bounds on this observable, especially due to
a strong tension with the $B_{u}\to\tau\nu$ bound.
