Below is the LTspice simulation of the circuit:
I used 0.5H inductance for the coil, which is in the ballpark for a small 12V relay.
As AK stated, the relay is apparently intended to be on continuously when the pulses are applied, although the average coil current seems a little low to pull in the relay.
That would make it a watch-dog circuit.

The value of R3 is limiting the current.
Better would be a push-pull or totem-pole driver (example circuit at bottom).

 

That's correct.
I tested the circuit with a very similar 12 volt relay with a 390 ohm coil, results are:
Average voltage across the coil was -7 volts, 18ma with a dissipation 126mw. Total average current draw was 42ma or 1watt dissipation
Compared to connecting relay in series with R2 and driving the FET directly.
Total current draw 30ma with a dissipation of .72 watt.
11.7 volts across relay coil, 30ma with a dissipation of 390mw or 3 times more then the previous result.

Can you measure the inductance of you relay coil? and share inductance and part number please?

 

Can you measure the inductance of you relay coil? and share inductance and part number please?

Impedance measure 3300 ohms at 1KHz, calculates to appx 0.53H.
Relay type: QIANJI.....JQC-3F-(T73) ......12 volt 390 ohm coil.
My measurements accurately matchup with crutschow's simulation using 0.5H for the coil with the same average Vout of -7 volts.