  Using a data sample collected with the BESIII detector operating at the
BEPCII storage ring at a center-of-mass energy of $\sqrt{s}=4.600$ GeV, we
search for the production of $e^{+}e^{-} \to \phi\chi_{c0,1,2}$ and the
charmonium-like state $Y(4140)$ in the radiative transition $e^{+}e^{-} \to
\gamma Y(4140)$ with $Y(4140)$ subsequently decaying into $\phi J/\psi$. The
processes $e^{+}e^{-} \to \phi\chi_{c1}$ and $\phi\chi_{c2}$ are observed for
the first time, each with a statistical significance of more than 10\sigma, and
the Born cross sections are measured to be $(4.2^{+1.7}_{-1.0}\pm 0.3)$ pb and
$(6.7^{+3.4}_{-1.7}\pm 0.5)$ pb, respectively, where the first uncertainties
are statistical and the second systematic. No significant signals are observed
for $e^{+}e^{-} \to \phi\chi_{c0}$ and $e^{+}e^{-} \to \gamma Y(4140)$ and
upper limits on the Born cross sections at $90\%$ confidence level are provided
at $\sqrt{s}=4.600$ GeV.
