There's no good solution to this problem. Overprinting white is not an option in offset printing. Trapping is impossible. There is simply no way to get tiny white type to knock out through a CMYK image and consistently look as crisp as it does on the computer screen. There are only two options: 1) Live with it. 2) Change it.
Could you screen back the background and overprint black?
Could you encase the white type in a picture box of 100% black, so that the text only knocks out through one solid color rather than four screened colors?
Designers often ask, how small can you make white knockout type and not worry about it? That's a hard question. 8 point Helvetica is pretty safe, but unless registration is close to perfect, you'll see haloes on the edge of the type, and the text will be considerably more tiring to read. It won't cross the line and become illegible mush, but depending on how lucky you are, it may or may not be user-friendly. On the other hand, 8 point Helvetica set as black overprint reads effortlessly, every time.