No, the Monthly Game is generated by your play on the Daily Games. Once midnight on the last day arrives it's then a new month, so it resets, and you start again.

Your picks cannot be carried over to a new month. It’s not something that can be compensated for either as not only is it free content, so it doesn’t have a monetary value, but we simply don’t know what you would have picked, you could have won something, but you may have won nothing.