IP routing on local machine ignores gateway

I have been looking at using an Internet gateway on a Pi 4. I have it setup to route default trafiic to my ADSL router, but with a static route set to route google trafic via an EE mobile router.

At first sight it appears to be working correctly. In fact the IP router is behaving properly…

But the local computer (another Pi 4) is making its own decision to change the routing for google traffic to go direct to the EE mobile router, ignoring the gateway setting.

The first connection goes via the gateway, but subsequent traffic by-passes the gateway.

I have found a work around for this.

Put the EE mobile router on a subnet that the local computer can not see.

Then all of the traffic continutes to go via the designated gateway. But only Google traffic then goes on via the EE mobile router after going to the designated gateway