  We consider the possibility of discovering a light pseudoscalar particle $a$,
which could the axion or an axion-like particle (ALP), interacting mainly with
quarks using the electron and photon scattering reaction chain $e + Z
\rightarrow e + \gamma + Z$; $\gamma + Z \rightarrow a + Z$ on nuclei in the
NA64 experiment at the CERN SPS. New bounds on the coupling strengths of the
axion/ALP with quarks and of pseudoscalar mesons $P = \pi, \eta, \eta'$ with
dark fermions are obtained by using existing upper bounds on invisible decay
modes of $P$s including those recently derived by NA64. We also study a
scenario when the $a$ plays the role of a messenger in the communication
between our world and the dark sector.
