What is the ruling on someone who intended to fast a supererogatory fast, then put some food in his mouth before Fajr and fell asleep, and woke up after the Adhan with the food still in his mouth, so he got rid of it and completed the fast?

What breaks a fast, whether it is a supererogatory or an obligatory fast, is the same. And what constitutes an excuse for breaking it is also the same. There is no difference between obligatory and supererogatory fasts regarding nullifiers, except that a supererogatory fast does not necessitate completion, nor does it require making up for it if one breaks it, according to the sounder opinion.