May one pour and/or mix gravy into mashed potatoes?^[1]

Yes.[2]

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[1] See Beis Yosef 321 in name of Teshuvos HaRambam brought also in M”A 321:28; Taz 321:14; Orchos Shabbos 6:13; M”A 321:28; Bach; Elya Raba 321:28; Chayeh Adam 20:21; Kaf Hachaim 321:134; Ketzos Hashulchan 130 footnote 12 regarding lentils, Chazon Ish 58:9; SSH”K 8:19; Shabbos Kehalacha 22:12-13; Piskeiy Teshuvos 321:23

[2] The reason: As the potatoes have already been kneaded with water when cooked and there are opinions which say that cooked foods which are now soft and easily mash-able do not contain the kneading prohibition and thus one may add liquid to them and knead the liquid into it even into a thick batter, using the regular methods. The reason for this is because the food is ready to be eaten in its current state, and hence adding water to it is not considered part of the Melacha of Lisha but rather as part of the eating process. [ibid] According to them one may thus add any liquid to mashed potatoes, or to cut pieces of cooked potatoes even if doing so makes it into a thick batter.

Other opinions who require Shinuiy:  See M”A 321:28; Bach; Elya Raba 321:28; Chayeh Adam 20:21; Kaf Hachaim 321:134

Opinion of Michaber:  The Michaber 321:19 permits to knead/mix a dish that came off the fire. It is however disputed amongst the Poskim here as to whether one must follow the kneading restrictions or not.

Opinion of Biur Halacha: The Biur Halacha 321:14 “Shema Yavo” states that that according to the opinion of Tosafos the Biblical kneading prohibition does not apply to cooked or baked foods. [This refers to the opinion of Tosfos Shabbos 18a in Rebbe Yossi which is part of the 2^nd opinion in Admur ibid] The reason for this is because the food is ready to be eaten in its current state, and hence adding water to it is not considered part of the Melacha of Lisha but rather as part of the eating process. Nevertheless, even according to this opinion it remains Rabbinically forbidden to do so, and thus may only be done with the irregularities listed by the Sages. Vetzaruch Iyun if this would apply likewise according to the opinion of the Rambam, which is the first opinion in Admur ibid. However certainly it would not apply according to the opinion of Rebbe Yehuda [2^nd opinion in Admur ibid], as he prohibits to knead roasted flour even with the irregularities, as brought in Shabbos 155b and Admur 321:16, and initially we are stringent like the opinion of Rebbe Yehuda. Practically, it seems that even according to the above novelty of the Biur Halacha he holds that cooked products must follow the regular kneading rules and hence may only be made into a thin mixture following the irregularities listed by the Sages.