Weak G-band (wGb) stars are a very peculiar class of red giants; they are almost devoided of carbon and often present mild lithium enrichment. Despite their very puzzling abundance patterns, very few detailed spectroscopic studies existed up to a few years ago, which prevented any clear understanding of the wGb phenomenon. We recently proposed the first consistent analysis of published data for a sample of 28 wGb stars and were able to identify them as descendants of early A-type to late B-type stars, although we were not able to conclude on their evolutionary status or the origin of their peculiar abundance pattern. Using new high-resolution spectra, we present the study of a new sample of wGb stars with the aim of homogeneously deriving their fundamental parameters and surface abundances for a selected set of chemical species that we use to improve our insight on this peculiar class of objects. We obtained high-resolution and high signal-to-noise spectra for 19 wGb stars in the southern and northern hemisphere that we used to perform consistent spectral synthesis to derive their fundamental parameters and metallicities, as well as the spectroscopic abundances for Li, C, ^12^C/^13^C, N, O, Na, Sr, and Ba. We also computed dedicated stellar evolution models that we used to determine the masses and to investigate the evolutionary status and chemical history of the stars in our sample. We confirm that the wGb stars are stars with initial masses in the range 3.2 to 4.2M_{sun}_. We suggest that a large fraction could be mildly evolved stars on the subgiant branch currently undergoing the first dredge-up, while a smaller number of stars are more probably in the core He burning phase at the clump. After analysing their abundance pattern, we confirm their strong nitrogen enrichment anti-correlated with large carbon depletion, characteristic of material fully processed through the CNO cycle to an extent not known in evolved intermediate-mass stars in the field and in open clusters. However, we demonstrate here that such a pattern is very unlikely owing to self-enrichment. In the light of the current observational constraints, no solid self-consistent pollution scenario can be presented either, leaving the wGb puzzle largely unsolved.