The respective contributions of gas accretion, galaxy interactions, and mergers to the mass assembly of galaxies, as well as the evolution of their molecular gas and star-formation activity are still not fully understood. In a recent work, a large sample of double-peak (DP) emission-line galaxies have been identified from the SDSS. While the two peaks could represent two kinematic components, they may be linked to the large bulges that their host galaxies tend to have. Star-forming DP galaxies display a central star-formation enhancement and have been discussed as compatible with a sequence of recent minor mergers. In order to probe merger-induced star-formation mechanisms, we conducted observations of the molecular-gas content of 35 star-forming DP galaxies in the upper part of the main sequence (MS) of star formation (SF) with the IRAM 30 m telescope. Including similar galaxies 0.3 dex above the MS and with existing molecular-gas observations from the literature, we finally obtained a sample of 52 such galaxies. We succeeded in fitting the same kinematic parameters to the optical ionised- and molecular-gas emission lines for ten (19%) galaxies. We find a central star-formation enhancement resulting most likely from a galaxy merger or galaxy interaction, which is indicated by an excess of gas extinction found in the centre. This SF is traced by radio continuum emissions at 150 MHz, 1.4 GHz, and 3 GHz, all three of which are linearly correlated in log with the CO luminosity with the same slope. The 52 DP galaxies are found to have a significantly larger amount of molecular gas and longer depletion times, and hence a lower star-formation efficiency, than the expected values at their distance of the MS. The large bulges in these galaxies might be stabilising the gas, hence reducing the SF efficiency. This is consistent with a scenario of minor mergers increasing the mass of bulges and driving gas to the centre. We also excluded the inwards-directed gas migration and central star-formation enhancement as the origin of a bar morphology. Hence, these 52 DP galaxies could be the result of recent minor mergers that funnelled molecular gas towards their centre, triggering SF, but with moderate efficiency.