How do fishes "breathe" while eating?
They can pump water over their gills with their opercular flaps (i.e., the structure that covers and protects the gills). This is assisted by the branchiostegal rays. Food would generally not interfere with this process. Additionally, gill rakers assist in keeping food in the mouth.

Some great references for questions like this:

Barton, M., &amp; Bond, C. E. (2007). Bond's biology of fishes. Thomson.

Moyle, P. B., &amp; Cech, J. J. (2004). Fishes: an introduction to ichthyology (Vol. 726). Upper Saddle River, NJ: Pearson Prentice Hall.