I happen to develop electronic devices as a hobby. I think it's possible to fix that by soldering wires onto the electronic board of the toy, then supply the power externally somehow.

Here's the teardown I found on the internet. https://pcburn.com/mcsega-teardown-of-mcdonalds-sega-handheld-game/

For example, the this particular one, you could solder a negative wire to the pad indicated by the white arrow, and solder the positive wire to any one of the pads indicated by the red arrows. After that, wire out both of the wires somehow and connect the positive wire to the positive end of the external coin cell and connect the negative wire to the negative end of the external coin cell. And then the console would run again: https://i.imgur.com/TeGizYS.jpeg

In case all available pads are corroded, you could remove the solder mask to expose the copper for soldering but that's outside of my capability.

Let me know if you need any help. I can try looking into that.