This indeed often happens and is a difficult question that got me thinking since I saw it for the first time. I didn't find it covered in any classical halacha seforim so we need to go back to the basics of brachot.
R Forst's sefer on brachot (pp. 133ff) halachically requires a bracha if the eating
gives benefit or satisfaction to the body -- benefit can be either the taste or the nutritional value (water is included here for the pleasure of quenching one's thirst)
is eaten in a natural manner -- food which is eaten in a manner other than the normal eating process is not considered eaten and does not require a bracha (e.g., intra-gastric feeding)
was eaten for normal eating purposes -- food eaten for tasting in small measure (less than a kazait) does not require a bracha according to a number of important authorities (Mishna Brura 210:13, Rambam cited in Orach Haim 210:2)
In addition, there is an important rule for brachot that safek brachot l'hakel. In case of doubt for brakhot de-rabannan, one abstains.
For all these reasons, it appears to me that eating remainders of food stuck in one's mouth doesn't require a bracha, as it doesn't give meaningful satisfaction (in taste or nutritional value), is not the natural way of eating and is in very small measure.
These reasons combined create a meaningful safek which should allow to swallow the food without bracha.
As always CYLOR