Trying to get two machines on different subnets to communicate

Hi, I'm sorry if this is a stupid question but I do not have much experience with networking. I am going to try and get my company to put me through a course (if anyone has any reccomendations?)

My issue is that I am trying to get two machines to communicate. They are on different subnets, but the subnet masks cannot be changed. The IP of either machine cannot be changed as well, as they have a lot of different devices connected to them.

I have a diagram of the current setup, which can be seen HERE.

There is a PC, connected to a switch, connected to a router, connected to another switch. Currently the settings are rubbish; the default gateway of the first PC points towards an I/O device which monitors the conditions of an exhaust tunnel.

I have been doing research to try and figure out how to get this to work and my proposed setup is HERE.

Is my understanding correct? When trying to connect to another IP, the device applies the subnet mask to the IP, then searches that set of IP's. If it does not find it, then it asks the device assigned to the default gateway if it can find it, and so on?

I know this is probably a stupid question, but I appreciate the help. I can't really use trial and error with this as the PC I have called "PC-A" is at the centre of operation. If it goes down, it has a great impact on everything. Thanks for any help!

tl;dr: Trying to get two machines on different subnets to talk to each other, without changing subnet mask. Machines are separated by a managed switch, and a router. Current config can be seen HERE. Proposed config can be seen HERE.