Help adding relay for thermostats

My nest thermostats are missing a common wire which means they’re not getting enough power to work properly. I actually didn’t realize this until an electrician told me about it (it hadn’t become a problem yet), but now it has and I brought him back in since he indicated he could do the work for $100 which seemed pretty reasonable. It turns out I need to add a new relay and now the price has gone way up so I’m going to try doing it myself.

I’m hoping someone can help me understand why I need a new relay and if this is something I can reasonably do myself. I’ve done some limited electrical work, but this is a bit advanced for my experience. If someone can help me better understand this system I imagine it’s something I could do myself, but perhaps it’s better left to a professional?

The white-coated wire here is a new wire since one of the two zones was a two wire and had to be replaced. There are actually three zones here, but I am only using nest thermostats on two of them at the moment (but it couldn’t hurt to future proof it). The two pre-existing ones are hot-water heat and the one I’m adding is steam heat. The brown-coated wires already have three wires. The green wire is being used as the common wire.

Here is the setup

High level view of the side – the brown and white wires are coming in from above. You can see the existing relay and transformer here.

relay

transformer and you can also see here there’s a Honeywell box that the white and red wires on the bottom of the relay run to – I’m guessing that’s for the hot water heat?

That brown wire that runs to the Honeywell part (not sure what that is?) then runs along with the white wire to the oil burner here:

Some additional images in case there’s anything I missed: