I want to add more time to this relay delay
As Jasen said, and as common sense kind of suggests: once you're changing the way a device works by multiple orders of magnitude, you typically have to redesign the device, rather than just change a couple components.
So, no, you cannot just exchange a couple of things. You need to exchange basically everything.
An analog timer circuit with 10 minutes hold is hard to build reliably, because you need very large capacitors, which means they are very inaccurate.
Considering the large size of this PCB and its components, you should be able to design a PCB that carries the same or a similar relay, and a microcontroller. Since you really don't need much from that microcontroller, that would put you in the 50ct region of component cost. So, full PCB + microcontroller + relay + passives would be &lt; 5€. That's a no-brainer. Design your own PCB, or use a timed relay that integrates that functionality from the start.