I think it's an old technique to help in fast turn off. When you have the FET ON, the voltage across the cap is 0V (R4 ensures that by shorting C3 and we're assuming some time for things to stabilize); then you switch the half-bridge to 0V (2N3904 -> off, 2N3906 -> on) which puts the capacitor in parallel with the FET's gate-source capacitance which is charged to Vcc, thus discharging it quickly to some point "in the middle" by having charge move from G-S to C3 without resistance (well, almost, there's the BJT + parasitics in the path). The faster the FET switch off, the higher the voltage transient obtained from the coil (dv = L (di/dt)).