10 minutes is a loooong time for a single RC timer stage. With a 1 meg resistor, the capacitor for 10 minutes would be around 500 to 700 uF, so large that its leakage current might be close to or even greater than the charging current, leading to serious timing errors that would change with temperature. Another approach is an oscillator with a much shorter period followed by a multi-stage divider, like the CD4060.
When power is applied the counter is zeroed out by R1 and C1, and then starts counting. Output Q14 is low at this time, holding Q1 on, and input DC power is applied to the output. R3 adjusts the oscillator frequency, and hence the timer period. After 8192 counts, Q14 goes high, the output is turned off, D1 inhibits further counting, and the circuit just sits there in the off state. Cycle power to restart. There is no information about the load current requirement, so the power MOSFET should be adjusted for that.
In this example, the "timing" capacitor is a small ceramic cap, 0.33 uF. This is 1,650 times smaller than the cap required for a standard 555 monostable circuit with a 10 minute period.
Edit - added LED to indicate timing activity.
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