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The aim of this work is to characterize HCN and CH$_3$OH emission from recent comets. We used the Onsala 20-m telescope to search for millimetre transitions of HCN towards a sample of 11 recent and mostly bright comets in the period December 2016 to November 2019. Also CH$_3$OH was searched for in two comets. The HCN sample includes the interstellar comet 2I/Borisov. For the short-period comet 46P/Wirtanen we could monitor the variation of HCN emission over a time span of about one month. We performed radiative transfer modelling for the observed molecular emission by also including time-dependent effects due to the outgassing of molecules. HCN was detected in 6 comets. Two of these are short-period comets and 4 of them are long-period. Six methanol transitions were detected in 46P/Wirtanen, enabling us to determine the gas kinetic temperature. From the observations, we determined the molecular production rates using time-dependent radiative transfer modelling. For 5 comets, we could determine that the HCN mixing ratios lie near 0.1% using contemporary water production rates, $Q_\mathrm{H_2O}$, taken from other studies. This HCN mixing ratio was also found typical in our monitoring observations of 46P/Wirtanen but here we notice deviations, on a daily time scale, up to 0.2% which could indicate short-time changes in the outgassing activity. From our radiative transfer modelling of cometary comae, we found that time-dependent effects on the HCN level populations are of the order 5-15% when $Q_\mathrm{H_2O}$ is around $2\times 10^{28}\,\mathrm{mol\, s^{-1}}$. The effects may be relatively stronger for comets with lower $Q_\mathrm{H_2O}$. The exact details of the time-dependent effects depend on the amount of neutral and electron collisions, radiative pumping, and molecular parameters such as the spontaneous rate coefficient.