  In this paper we produce new, optimal, regularity results for the solutions
to $p$-Poisson equations. We argue through a delicate approximation method,
under a smallness regime for the exponent $p$, that imports information from a
limiting profile driven by the Laplace operator. Our arguments contain a
novelty of technical interest, namely a sequential stability result; it
connects the solutions to $p$-Poisson equations with harmonic functions,
yielding improved regularity for the former. Our findings relate a smallness
regime with improved $\mathcal{C}^{1,1-}$-estimates in the presence of
$L^\infty$-source terms.
