  We study the decays $B_{c,u,d}\to X(3872)P$ in the perturbative QCD (PQCD)
approach, where the puzzling resonance $X(3872)$ is involved and $P$ represents
a light pseudoscalar meson $K$ and $\pi$. Assuming the $X(3872)$ as a $1^{++}$
charmonium state, we find the following results: (a) The branching ratios for
the decays $B^+_c\to X(3872)\pi^+$ and $B^+_c\to X(3872) K^+$ agree with the
results predicted by the covariant light-front approach within errors, but are
larger than those given by the generalized factorization approach; (b) The
branching ratio for the decay $B^+\to X(3872)K^+$ is predicted as
$(3.8^{+1.1}_{-1.0})\times10^{-4}$, which is smaller than the previous PQCD
calculation result, but still slightly larger than the upper limits set by
Belle and BaBar. So we suggest that the decays $B^{0,+}\to X(3872)K^{0,+}$
should be precisely measured by the running LHCb and Belle II experiments,
which is very helpful to probe the inner structure of the $X(3872)$; (c)
Compared with the decays $B_{u,d}\to X(3872)K$, the decays $B_{u,d}\to
X(3872)\pi$ have much smaller branching ratios, which drop to as low as
$10^{-6}$; (d) The direct CP violations for these considered decays are very
small, only $10^{-3}\sim 10^{-2}$, because the penguin contributions are loop
suppressed compared with the tree contributions. Testing the results for the
branching ratios and the CP violations including the implicit $SU(3)$ and
isospin symmetries in these decays by experiments is helpful to probe the
nature of the $X(3872)$.
