Aims: We present the abundance analysis of 12 planetary nebulae ionized by [WC]-type stars and weak-emission-line stars (wels) obtained from high-resolution spectrophotometric data. Our main aims are to determine the chemical composition of the nebulae and to study the behaviour of the abundance discrepancy problem (ADF) in this type of planetary nebulae. Methods: The detection of a large number of optical recombination lines (ORLs) and collisionally excited lines (CELs) from different ions (O+, O++, C++, C+3 and Ne++) were presented previously. Most of the ORLs were reported for the first time in these PNe, which increased the sample of PNe with detected faint ORLs. Ionic abundances were determined from the available CELs and ORLs, using previously determined physical conditions. Based on these two sets of ionic abundances, we derived the total chemical abundances in the nebulae using suitable ionization correction factors (when available). Results: In spite of the [WC] nature of the central stars, moderate ADF(O++) in the range from 1.2 to 4 were found for all the objects. We found that when the quality of the spectra is high enough, the ORLs O++/H+ abundance ratios obtained from different multiplets excited mainly by recombination are very similar. Possible dependence of ADFs on some nebular characteristics such as surface brightness and nebular diameter were analysed, but we found no correlation. Abundances derived from CELs were corrected by determining the t2 temperature fluctuation parameter. O abundances for PNe, derived from ORLs, are in general higher than the solar abundance. We derived the C/O ratio from ORLs and N/O and α-element/O ratios from CELs and found that these PNe are, on average, richer in N and C than the average of the large PN samples. About half of our sample is C-rich (C/O > 1). The growth of α-elements is correlated with the O abundance. Comparing the N/O and C /O ratios with those derived from stellar evolution models, we estimate that about half of our PNe have progenitors with initial masses similar to or larger than 4 M⊙. No correlation was found between the stellar [WC] type and the nebular chemical abundances. A rough O abundance gradient computed for our limited PN sample, compared with the gradient obtained for H ii regions, shows that there is a large dispersion in estimates of the PNe O abundance for a given Galactocentric distance. The PN gradient is flatter than that for H ii regions and at the solar distance and farther out, the PNe have a higher O abundance than H ii regions, similarly to what is found in other spiral galaxies. This fact has no convincing explanation so far. Based on data obtained at Las Campanas Observatory, Carnegie Institution.