  We consider the production of electroweak $Z$ or $W^\pm$ bosons associated
with $J/\psi$ mesons at the LHC conditions. Our attention is focused on new
partonic subprocesses which yet have never been considered in the literature,
namely, the charmed or strange quark excitation subprocesses followed by the
charmed quark fragmentation $c\to J/\psi+c$. Additionally we take into account
the effects of multiple quark and gluon radiation in the initial and final
states. We find that the contributions from the new mechanisms are important
and significantly reduce the gap between the theoretical and experimental
results on the $J/\psi + Z$ and $J/\psi + W^\pm$ production cross sections.
