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Let $\sigma_a^{(N)}(n)=\sum_{d^{N}|n}d^a$. An explicit transformation isobtained for the generalized Lambert series$\sum_{n=1}^{\infty}\sigma_{a}^{(N)}(n)e^{-ny}$ for Re$(a)>-1$ using therecently established Vorono\"i summation formula for $\sigma_a^{(N)}(n)$, andis extended to a wider region by analytic continuation. For $N=1$, this Lambertseries plays an important role in string theory scattering amplitudes as can beseen in the recent work of Dorigoni and Kleinschmidt. These transformationsexhibit several identities - a new generalization of Ramanujan's formula for$\zeta(2m+1)$, an identity associated with extended higher Herglotz functions,generalized Dedekind eta-transformation, Wigert's transformation etc., all ofwhich are derived in this paper, thus leading to their uniform proofs. Aspecial case of one of these explicit transformations naturally leads us toconsider generalized power partitions with ``$n^{2N-1}$ copies of $n^{N}$''.Asymptotic expansion of their generating function as $q\to1^{-}$ is alsoderived which generalizes Wright's result on the plane partition generatingfunction. In order to obtain these transformations, several new intermediateresults are required, for example, a new reduction formula for Meijer$G$-function and an almost closed-form evaluation of $\left.\frac{\partialE_{2N, \beta}(z^{2N})}{\partial\beta}\right|_{\beta=1}$, where $E_{\alpha,\beta}(z)$ is a two-variable Mittag-Leffler function.

Explicit transformations for generalized Lambert series associated with the divisor function $\sigma_{a}^{(N)}(n)$ and their applications