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We demonstrate a possibility of the creation of stable optical solitons combining one continuous and one discrete coordinate, with embedded vorticity, in an array of planar waveguides with intrinsic cubic-quintic nonlinearity. The same system may be realized in terms of the spatiotemporal light propagation in an array of tunnel-coupled optical fibers with the cubic-quintic nonlinearity. In contrast with zero-vorticity states, semidiscrete vortex solitons do not exist without the quintic term in the nonlinearity. Two types of the solitons, \emph{viz.}, intersite- and onsite-centered ones (IC and OC, respectively), with even and odd numbers $N$ of actually excited sites in the discrete direction, are identified. We consider the modes carrying the embedded vorticity $S=1$ and $2$. In accordance with their symmetry, the vortex solitons of the OC type exhibit an intrinsic core, while the IC solitons with a small $N$ may have a coreless structure. Facilitating their creation in the experiment, the modes reported in the present work may be much more compact states than their counterparts considered in other systems, and they feature strong anisotropy. They can be set in motion in the discrete direction, provided that the coupling constant exceeds a certain minimum value. Collisions between moving vortex solitons are considered too.