  We study one-loop effects induced by an anomalous Higgs trilinear coupling on
total and differential rates for the $H\to 4\ell$ decay and some of the main
single-Higgs production channels at the LHC, namely, VBF, $VH$, $t\bar tH$ and
$tHj$. Our results are based on a public code that calculates these effects by
simply reweighting samples of Standard-Model-like events for a given production
channel. For $VH$ and $t\bar tH$ production, where differential effects are
particularly relevant, we include Standard Model electroweak corrections, which
have similar sizes but different kinematic dependences. Finally, we study the
sensitivity of future LHC runs to determine the trilinear coupling via
inclusive and differential measurements, considering also the case where the
Higgs couplings to vector bosons and the top quark is affected by new physics.
We find that the constraints on the couplings and the relevance of differential
distributions critically depend on the expected experimental and theoretical
uncertainties.
