Relay switching off when under load (spot welder)
First of all, the RESET pin in your schematic is not connected to the supply voltage Vcc, which may cause spurious resets.

Secondly, the supply voltage is likely to drop during welding, which will lower the reset voltage threshold (in your schematic it's always 2/3 of Vcc); if the reset voltage threshold drops below the voltage on your capacitor, the 555 will turn off. I suggest that you power the 555 through a voltage regulator such as an LM3805/кр142ен5/кр1158ен5 and substitute a BJT for your MOSFET, as BJTs only need 0.6 V to turn on.

Note that if things happen quickly, you cannot measure the true voltage drop using a multimeter, as it always gives you the average voltage over a certain period.