A "mulligan" is a second chance to enter your daily score if you missed your chance to enter the first time around. You may choose to use one of your mulligans for a second chance to enter that score. A mulligan is not an opportunity to go back and give yourself a perfect score, it's just an opportunity to log a score for a day that has passed.
Click on the number next to the word "mulligan" just above your rainbow chart, or check your notifications and accept the offer to use a mulligan for a missed day.
Once you run out of mulligans you will no longer see any missed days since you will no longer be able to access them for scoring.
There are no penalties for using a mulligan. All players begin the game with 3 mulligans.