  A study of $B_c^+\to K^+K^-\pi^+$ decays is performed for the first time
using data corresponding to an integrated luminosity of 3.0 $\mathrm{fb}^{-1}$
collected by the LHCb experiment in $pp$ collisions at centre-of-mass energies
of $7$ and $8$ TeV. Evidence for the decay $B_c^+\to\chi_{c0}(\to K^+K^-)\pi^+$
is reported with a significance of 4.0 standard deviations, resulting in the
measurement of
$\frac{\sigma(B_c^+)}{\sigma(B^+)}\times\mathcal{B}(B_c^+\to\chi_{c0}\pi^+)$ to
be $(9.8^{+3.4}_{-3.0}(\mathrm{stat})\pm 0.8(\mathrm{syst}))\times 10^{-6}$.
Here $\mathcal{B}$ denotes a branching fraction while $\sigma(B_c^+)$ and
$\sigma(B^+)$ are the production cross-sections for $B_c^+$ and $B^+$ mesons.
An indication of $\bar b c$ weak annihilation is found for the region
$m(K^-\pi^+)<1.834\mathrm{\,Ge\kern -0.1em V\!/}c^2$, with a significance of
2.4 standard deviations.
