  The excellent experimental prospects to measure the invisible mass spectrum
of the $K^+\to\pi^+\nu\bar\nu$ decay opens a new path to test generalised
quark--neutrino interactions with flavour changing $s\to d$ transitions and as
such to novel probes of Physics beyond the Standard Model. Such signals can be
a consequence of new lepton-number violating or lepton-number conserving
interactions, with their interpretations depending on the Majorana versus Dirac
nature of the neutrinos. Furthermore, the possible existence of new massive
sterile neutrinos can be tested via their distinctive imprints in the invariant
mass spectrum. Within the model-independent framework of the weak effective
theory at dimension-six, we study the New Physics effects of Majorana and Dirac
neutrinos on the differential distribution of $K^+\rightarrow \pi^+\nu\bar\nu$
allowing for lepton-number violating interactions and potential new sterile
neutrinos. We determine the current and expected future sensitivity on the
corresponding $\Delta S=1$ neutral-current Wilson coefficients using the
distribution measured by the NA62 collaboration and accounting for expected
improvements based on the HIKE experiment. We present single-operator fits and
also determine correlations among different type of operators. Even though we
focus on $s\to d\nu\nu$ transitions, the operator bases for Majorana and Dirac
and the classification of lepton-number-violating/conserving interactions is
applicable also for the study of $b\to s/d\nu\nu$ and $c\to u\nu\nu$
transitions relevant in current phenomenology.
