What is the the purpose of the capacitor in this relay circuit?
The data sheet for that relay -
https://www.mouser.co.uk/datasheet/2/307/en-g6k-348800.pdf

Shows that it is a latching relay.  The capacitor will allow a pulse of current to the relay coil to activate the latch, but will then block the DC.  Once latched no further current will be drawn.
Without more information as to what the circuit is supposed to be doing, it is not possibly to say why it is designed like this.