If the current in the circuit for heating the filament is increased, the cut-off wavelength
The electrons are emitted by thermionic emission and then accelerated towards the target by using a different circuit. (See the diagram and notice the two circuits). When current is increased in the heating circuit, it will only allow the out coming electrons to cross the work-function fast, once electrons are emitted they are completely in the influence of the second circuit. The potential difference between the cathode and anode provides energy to the electron.
Hence changing the current will have no effect on electron's energy and hence on the cut-off wavelength.