Q. On Shabbos the congregants can read with difficulty, without the lights. Can one ask a Goy to turn ?on the lights

A. Shulchan Aruch (O.H. 307: 5) and Mishna Berura (ibid. 24) prohibit instructing a Gentile to perform a Biblically proscribed melacha such as turning on an electric light, even when needed for a mitzvah such as davening or learning Torah.
Noda Beyehuda (M.K. O.H. 33, see also M.T. 33) rules stringently in regards to those who request Gentiles to light candles before Neilah on Yom Kipur. .He advises to rather avoid reciting the additional piyutim. See also Heishiv Moshe (10) .
Even if the gentile was not asked but noticed on his own that the light is needed and turned it on by his own desire, it cannot be used since it was lighted for the benefit of the Jewish congregants. (ibid. 276: 1.)
However, in this last case if one could read, although with difficulty without the added light, it would be permitted to use that light. If not at all, then the light cannot be used. (ibid. 276: 4)

Rabbi A. Bartfeld as revised by Horav Shlomo Miller Shlit”a