The slow neutron capture process in massive stars (weak s process) produces most of the s-process isotopes between iron and strontium. Neutrons are provided by the ²²Ne(α,n)²⁵Mg reaction, which is activated at the end of the convective He-burning core and in the subsequent convective C-burning shell. The s-process-rich material in the supernova ejecta carries the signature of these two phases. In the past years, new measurements of neutron capture cross sections of isotopes beyond iron significantly changed the predicted weak s-process distribution. The reason is that the variation of the Maxwellian-averaged cross sections (MACS) is propagated to heavier isotopes along the s path. In the light of these results, we present updated nucleosynthesis calculations for a 25 M _☉ star of Population I (solar metallicity) in convective He-burning core and convective C-burning shell conditions. In comparison with previous simulations based on the Bao et al. compilation, the new measurement of neutron capture cross sections leads to an increase of s-process yields from nickel up to selenium. The variation of the cross section of one isotope along the s-process path is propagated to heavier isotopes, where the propagation efficiency is higher for low cross sections. New ⁷⁴Ge, ⁷⁵As, and ⁷⁸Se MACS result in a higher production of germanium, arsenic, and selenium, thereby reducing the s-process yields of heavier elements by propagation. Results are reported for the He core and for the C shell. In shell C-burning, the s-process nucleosynthesis is more uncertain than in the He core, due to higher MACS uncertainties at higher temperatures. We also analyze the impact of using the new lower solar abundances for CNO isotopes on the s-process predictions, where CNO is the source of ²²Ne, and we show that beyond Zn this is affecting the s-process yields more than nuclear or stellar model uncertainties considered in this paper. In particular, using the new updated initial composition, we obtain a high s-process production (overproduction higher than ¹⁶O, ~100) for Cu, Ga, Ge, and As. Using the older abundances by Anders & Grevesse, also Se, Br, Kr, and Rb are efficiently produced. Our results have important implications in explaining the origin of copper in the solar abundance distribution, pointing to a prevailing contribution from the weak s-process in agreement with spectroscopic observations and Galactic chemical evolution calculations. Because of the improvement due to the new MACS for nickel and copper isotopes, the nucleosynthesis of copper is less affected by nuclear uncertainties compared to heavier s-process elements. An experimental determination of the ⁶³Ni MACS is required for a further improvement of the abundance prediction of copper. The available spectroscopic observations of germanium and gallium in stars are also discussed, where most of the cosmic abundances of these elements derives from the s-process in massive stars.