During Aseret Yeme Teshuva the Shulhan Aruch (Orah Haim 603:1) states that one who normally eats bread products baked by non-Jews (Pat Akum/Nochri) should eat food baked by Jews (Pat Israel). Many products nowadays, such as cookies or bread, may have a Kosher certification but are not Pat Israel. Although it may seem like one is accepting an insincere and unrealistic stringency upon oneself, doing so is symbolic of one’s deep desire to be at such a spiritual level. Therefore, even if one cannot maintain such a stringency after Yom Kippur, one is showing Hashem what one’s ultimate ambition is.
The Magen Avraham explains that in previous generations, people would be strict about eating food that was ritually pure, and the Ben Ish Hai followed this practice. Nowadays this practice is uncommon.
The Arizal speaks of fasting during Aseret Yeme Teshuva as another stringency. Nowadays, our physical and spiritual strengths are diminished and fasting on Tzom Gedalia suffices. The study of Torah, however, is another means to achieve Teshuva in place of fasting and therefore one should make an effort to increase one’s learning.
Summary: One should make an effort to eat Pat Israel during Aseret Yeme Teshuva.