We analyze the stability of a Taylor-Couette flow under the imposition of a weak axial flow in the case of a very short cylinder with a narrow annulus gap. We consider an incompressible viscous fluid contained in the narrow gap between two concentric short cylinders, in which the inner cylinder rotates with constant angular velocity. The caps of the cylinders have narrow tubes conically tapering to very narrow slits, allowing an axial flow along the surface of the inner cylinder. The approximated solution for the Taylor-Couette flow for short cylinders was found and used for the stability analysis instead of the precise but bulky solution. The sensitivity of the Taylor-Couette flow to small perturbations and to weak axial flow was studied. We demonstrate that perturbations coming from the axial flow cause the propagation of dispersive waves in the Taylor-Couette flow. While in long cylinders the presence of an axial flow leads to the breaking of axial symmetry, in small cylinders it leads to the breaking of mirror symmetry. The coexistence of a rotation and an axial flow requires that, in addition to the energy and the angular momentum of the flow, the helicity must also be studied. The approximated form for the helicity formula in the case of short cylinders was derived. We found that the axial flow stabilizes the Taylor-Couette flow. The supercritical flow includes a rich variety of vortical structures, including a symmetric pair of Taylor vortices, an anomalous single vortex and quasiperiodic oscillating vortices. Pattern formation was studied at large for rated ranges of azimuthal and axial Reynolds numbers. A region where three branches of different states occur was localized. Numerical simulations in 3-D and in the axisymmetrical case of the model flow are presented, which illustrate the instabilities analyzed.