  We investigate conditions under which kink magnetohydrodynamic waves
propagating along photospheric uniformly twisted flux tubes with axial mass
flows become unstable as a consequence of the Kelvin-Helmholtz instability. We
employed the dispersion relations of kink waves derived from the linearised
magnetohydrodynamic equations. We assumed real wave numbers and complex angular
wave frequencies, namely complex wave phase velocities. The dispersion
relations were solved numerically at fixed input parameters and several mass
flow velocities. We show that the stability of the waves depends upon four
parameters, the density contrast between the flux tube and its environment, the
ratio of the background magnetic fields in the two media, the twist of the
magnetic field lines inside the tube, and the value of the Alfven-Mach number
(the ratio of the jet velocity to Alfv\'en speed inside the flux tube). At
certain densities and magnetic field twists, an instability of the
Kelvin-Helmholtz type of kink (m = 1) mode can arise if the Alfven-Mach number
exceeds a critical value. The observed mass flows may trigger the
Kelvin-Helmholtz instability of the kink (m = 1) mode in weakly twisted
photospheric magnetic flux tubes at critical Alfven-Mach numbers lower that
those in untwisted tubes if the magnetic field twist lies in the range
0.36--0.4 and the flow speed exceeds a critical value. A weak external magnetic
field (with a ratio to the magnetic field inside the tube in the range
0.1--0.5) slightly increases that critical value.
