Spin currents are paramount to manipulate the magnetization of ferromagnetic elements in spin-based memory, logic and microwave devices, and to induce spin polarization in non-magnetic materials. A unique approach to create spin currents employs thermal gradients and heat flow. Here we demonstrate that a thermal spin current can be tuned conveniently by a voltage. In magnetic tunnel contacts to semiconductors (silicon and germanium), it is shown that a modest voltage (similar to 200 mV) changes the thermal spin current induced by Seebeck spin tunnelling by a factor of five, because it modifies the relevant tunnelling states and thereby the spin-dependent thermoelectric parameters. The magnitude and direction of the spin current is also modulated by combining electrical and thermal spin currents with equal or opposite sign. The results demonstrate that spin-dependent thermoelectric properties away from the Fermi energy are accessible, and open the way towards tailoring thermal spin currents and torques by voltage, rather than material design.