Bremsstrahlung emission remains a cornerstone process in the theoretical characterization of electron dynamics in diverse high-energy environments. In particular, the accurate description of thin-target electron-ion bremsstrahlung in the presence of high-Z species requires careful treatment of atomic screening effects, especially when atoms are partially ionised. We present a unified formalism based on a multi-Yukawa representation of the screened atomic potential, enabling the calculation of bremsstrahlung cross-sections for arbitrary nuclear charge and ionisation state. This framework extends prior treatments (see H. Bethe and W. Heitler, 1934; G. Roche et al., 1972) of neutral atoms to encompass a broader class of systems. In particular, it applies to partially ionised high-Z elements typically found in plasma-facing components and disruption mitigation schemes in magnetically confined fusion devices. The method offers analytical tractability in both non-relativistic and ultra-relativistic regimes, supporting a consistent and accurate description of bremsstrahlung across various energy scales and screening conditions. Numerical results have been produced using a high-performance implementation of the model in MATLAB, optimised for efficient evaluation across large parameter spaces. **Funding**\ This work has been carried out within the framework of the EUROfusion Consortium, via the Euratom Research and Training Programme (Grant Agreement No 101052200 - EUROfusion) and funded by the Swiss State Secretariat for Education, Research and Innovation (SERI). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union, the European Commission, or SERI. Neither the European Union nor the European Commission nor SERI can be held responsible for them.