4V is barely enough to start turning on the MOSFET.
The circuit is not correct. The MOSFET needs a larger gate voltage to properly operate so you would need to add some gain.
The easiest is just to use a bipolar transistor in place of the MOSFET with the emitter to ground, a resistor from the base to you oscillator output, and a collector resistor to the +12V. You cannot apply a unipolar signal to a transformer since that creates net DC current which will saturate the core. So you must connect the collector output to the transformer input through a large capacitor (to block the DC).
How much power do you need out of the transformer? If you want significant power then you would need to use a complementary transistor driver instead of the one transistor with a collector resistor.