Repeating "a un gioco" is not useful in this case since the verb "giocare" already exhaustively tell us which action are we doing; moreover this shorter translation is already used in Telegram X for Android (https://translations.telegram.org/it/android_x/private_chats/SendingGame).

In addition to what I said for the same string in other clients, here "..." should also be removed since they create inconsistency with all other official clients

I second this.