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Abstract In this work, we investigate the complete spectroscopy of the $$B_c$$ B c mesons, with a special focus on the consideration of the unquenched effects. To account for such effects, we employ the modified Godfrey–Isgur model and introduce a screening potential. The resulting mass spectrum of the concerned higher $$B_c$$ B c states is then presented, showing significant deviations after considering the unquenched effects. This emphasizes the importance of considering the unquenched effects when studying of the higher $$B_c$$ B c mesons. Furthermore, we determine the corresponding spatial wave functions of these $$B_c$$ B c mesons, which have practical applications in subsequent studies of their decays. These decays include two-body Okuba–Zweig–Iizuka allowed strong decays, dipion transitions between $$B_c$$ B c mesons, radiative decays, and some typical weak decays. With the ongoing high-luminosity upgrade of the Large Hadron Collider, we expect the discovery of additional $$B_c$$ B c states in the near future. The knowledge gained from the mass spectrum and the different decay modes will undoubtedly provide valuable insights for future experimental explorations of these higher $$B_c$$ B c mesons.