  In this work, we investigate the spectroscopy of higher $B_c$ 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 $B_c$
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 $B_c$ mesons. Furthermore, we
determine the corresponding spatial wave functions of these $B_c$ 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 $B_c$ 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 $B_c$ 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 $B_c$
mesons.
