B supergiants (BSGs) are evolved stars with effective temperatures between 10 to 30 kK and are important to understand massive star evolution. Located on the edge of the line-driven wind regime, the study of their atmospheres is helpful to understand phenomena such as the bi-stability jump. Key UV features of their spectra have so far not been reproduced by models for types later than B1. Here, we aim to remedy this situation via spectral analysis that accounts for wind clumping and X-rays. In addition, we investigate the evolutionary status of our sample stars based on the obtained stellar parameters. We determined parameters via quantitative spectroscopy using CMFGEN and PoWR codes. The models were compared to UV and optical data of four BSGs: HD206165, HD198478, HD53138, and HD164353. We also study the evolutionary status of our sample using GENEC and MESA tracks. When including clumping and X-rays, we find good agreements between synthetic and observed spectra for our sample stars. For the first time, we reproduced key lines in the UV. For that, we require a moderately clumped wind (f_infty > ~0.5). We also infer relative X-ray luminosities of ~10^-7.5 to 10^-8 – lower than the typical ratio of 10^-7. Moreover, we find a possible mismatch between evolutionary and spectroscopic masses, which could be related to the mass-discrepancy problem present in other OB stars. Our results provide evidence that X-rays and clumping are needed to describe the winds of cool BSGs. However, their winds seem less structured than in earlier type stars. This aligns with observational X-rays and clumping constraints as well as recent hydrodynamical simulations. The BSGs’ evolutionary status appears diverse: some objects are potentially post-red supergiants or merger products. The wind parameters provide evidence for a moderate mass-loss rate increase around the bi-stability jump. Abstract abridged