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Someone on the Phaser's Discord channel suggested the following : 

two options 
1) you can call a normal JS function that creates and starts the game -- this would basically just be something like
function startMiniGame() { return new Phaser.Game(...) } 
and when it's over you would just destroy the game / make sure you drop all references and let it get garbage collected 

2) the other option would be to start the game in a hidden div and when it's time to start it your function would just call update a boolean that your game tracks and start playing; it'd look something like 

function startMiniGame() { document.getElementById('phaser-game').style.display = 'absolute'; window.gameRef.miniGameEnabled = true; }
 the actual details will vary wildly and I don't remember CE well at all so none of that sample code can be used as is but those are probably your two best approaches

In the two options I don't understand where and how  to put the mini-game code. What about a mini game with multiple files ?