Fruit fly type bugs always appear in the kitchen when the weather heats up
These small flies are very hard to keep out because they are so small, so I think looking for what is attracting them is a better long term solution.  This is a guess, but I think that using bleach in the drain is probably masking the scent of what is truly attracting them.  A small flying insect making it's way through a water-filled P-Trap under the sink is unlikely, and unless you also have a sewer smell coming from the sink, the P-trap is probably working correctly.

Look for dead portions of houseplants, fruit or veggies that have fallen behind a cabinet or the fridge or in the cracks next to the stove, etc.  Clean, clean, clean.  Pull out the stove and the fridge and clean.  Empty out the cabinet under the sink and clean that very well.  Pull out everything from the pantry and get rid of the expired stuff, and clean.

I'm in Texas as well and anytime we have had tiny little flies at our house or my mother in law's, its been from potatoes that were forgotten or something else that was leaking or rotting out of site that we didn't know about.