  The nature of the gas in CO-rich debris discs remains poorly understood, as
it could either be a remnant from the earlier Class II phase or of secondary
origin, driven by the destruction of icy planetesimals. The aim of this paper
was to elucidate the origin of the gas content in the debris discs via various
simple molecules that are often detected in the less-evolved Class II discs. We
present millimetre molecular line observations of nine circumstellar discs
around A-type stars: four CO-rich debris discs (HD 21997, HD 121617, HD 131488,
HD 131835) and five old Herbig Ae protoplanetary discs (HD 139614, HD 141569,
HD 142666, HD 145718, HD 100453). The sources were observed with the Atacama
Large Millimeter/submillimeter Array (ALMA) in Bands 5 and 6 with 1-2''
resolution. The Herbig Ae discs are detected in the CO isotopologues, CN, HCN,
HCO$^+$, C2H, and CS lines. In contrast, only CO isotopologues are detected in
the debris discs, showing a similar amount of CO to that found in the Herbig Ae
protoplanetary discs. Using chemical and radiative transfer modelling, we show
that the abundances of molecules other than CO in debris discs are expected to
be very low. We consider multiple sets of initial elemental abundances with
various degrees of H2 depletion. We find that the HCO+ lines should be the
second brightest after the CO lines and that their intensities strongly depend
on the overall CO/H$_2$ ratio of the gas. However, even in the ISM-like
scenario, the simulated HCO+ emission remains weak as required by our
non-detections.
