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Similar to my thoughts...
Change the supply to 12V (no change)
C1 is initially a short so the FET turns on.
C1 charges so the FET's gate (like transistor base) drops below the required (say) 3V to 5V between Gate and Source (ground) that turn the FET on.
R1 is to discharge the cap between power cycles (off to on).
D1 is to prevent the FET's Gate going -ve due to the charged cap (and power removal).
PPS - components not critical, though I'd use a IN400x for the relay's D2 (not the 1N4148), and may as well use the same for D1.
R2 x C1 set the "on" time.
R1 x C1 is the reset time (minimum "off" time required for another power-on 3-second-on. Else R1 could be omitted if SWitch is SPDT with "off" to GND - but then make sure D1 is fitted.