[mysql] How add unique key to existing table (with non uniques rows)

You can do as yAnTar advised

ALTER TABLE TABLE_NAME ADD Id INT AUTO_INCREMENT PRIMARY KEY

OR

You can add a constraint

ALTER TABLE TABLE_NAME ADD CONSTRAINT constr_ID UNIQUE (user_id, game_id, date, time)

But I think to not lose your existing data, you can add an indentity column and then make a composite key.