  The CP nature of the Higgs coupling to top quarks is addressed in this paper,
in single charged lepton final states of $t\bar{t}h$ events produced in
proton-proton collisions at the LHC. Pure scalar ($h=H$) and pseudo-scalar
($h=A$) Higgs boson signal events, generated with MadGraph5_aMC@NLO, are fully
reconstructed using a kinematic fit. Angular distributions of the decay
products, as well as CP-sensitive asymmetries, are exploited to separate and
gain sensitivity to possible pseudo-scalar components of the Higgs boson and
reduce the contribution from the dominant irreducible background
$t\bar{t}b\bar{b}$. Significant differences are found between the pure CP-even
and -odd signal hypotheses as well as with respect to the Standard Model
background, in particular the $t\bar{t} b\bar{b}$ contribution. Such
differences survive the event reconstruction, allowing to define optimal
observables to extract the Higgs couplings parameters from a global fit. A
dedicated analysis is applied to efficiently identify signal events and reject
as much as possible the expected Standard Model background. The results
obtained are compared with a similar analysis in the dilepton channel. We show
that the single lepton channel is more promising overall and can be used in
combination to study the CP nature of the Higgs coupling to top quarks.
