Tractate Peah, Chapter 3
We do not hire Non-Jewish workers [to harvest the fields],2 because they are not well versed3 in [leaving] Leket (fallen stalks), [Shikcha (forgotten sheaves), and Peah (corners of the field)].4
מסכת פאה פרק ג
אֵין שׂוֹכְרִין פּוֹעֲלֵי גוֹיִם לְפִי שֶׁאֵינן פְּקִיעִין בַּלֶּקֶט.
The Tosefta states a new law regarding gifts to the poor. It is not related to any Mishna.
As long as the Non-Jews harvest the produce for a Jewish owner of the field the produce remains obligated in all of the gifts to the poor, since they are acting as the messengers of the Jewish owner, and it is considered as if the owner himself is doing the harvesting. Unlike if the Non-Jews harvest for themselves, even if they are stealing produce of a Jew from his field, then all of the harvested produce in that field is exempt from all gifts to the poor, since it was not harvested for a Jew. See Mishna Peah 2:7 and Talmud Yerushalmi (Peah 2:5, Daf 13b).
The word פְּקִיעַ (Pekiya), or as sometimes it is spelled פָּקִיעַ (Pakiya), is a variant of the word בקי (Baki) or בקיא (Bakiya), which means “to be an expert” or “well versed” in something. According to Michael Sokoloff it is a specific variant in the Babylonian Aramaic dialect. See Michael Sokoloff, “A Dictionary of Jewish Babylonian Aramaic,” Bar Ilan University Press, 2002, p. 230, entry בקיע. However, this is not so clear to me as this word appears in multiple Palestinian sources, such as this Tosefta, Tosefta Pesachim 2:2 Vienna Manuscript and Geniza Fragment G2 TS E2 139, Tosefta Yoma 2:5-6 Vienna Manuscript and Geniza Fragment, Bologna State Archive 375, and Tosefta Ohalot 17:12 Vienna Manuscript. It is not really clear why these Toseftot would use a Babylonian Aramaic word and not the more common Palestinian word בקיא (Bakiya).
The Tosefta uses Leket as an example, but the same law would apply to Shikcha and Peah as well as is apparent from the discussion in Talmud Yerushalmi (Peah 2:5, Daf 13b).
In Talmudic times the Jews had a general distrust of the Non-Jews, especially in religious matters, so it was not even an option to consider for the Jewish owner of the field to give explicit instructions to his Non-Jewish workers to leave gifts to the poor, since they were not trusted to follow them. Therefore the Rabbis simply forbade Jewish field owners to hire Non-Jewish workers to do the harvest.