And as I have Electrum already installed, I was trying to avoid to install more stuff.
If that's the case then you can use Electrum without installing any additional software. I thought that you necessarily wanted to use your c-lightning node. I wrote a
walkthrough for Lighting Network support in Electrum. You might find it useful.
I doubt I can connect the game directly to my LN node and be able to receive the sats directly to my node in some way.
You can use your Lightning Network node to receive payments but you need to open a channel first and make room of incoming funds by spending some coins over the LN. The same applies to Electrum which might be easier for you to use.