Hungarian and Finnish are distantly related, but that's it. Turkish is unrelated to them and to Mongolian, which is also unrelated to them. The Altaic language-family hypothesis would have Mongolic languages related to Turkic ones, but that is not bolstered by good evidence and is not accepted by the linguistic establishment in countries where linguists actually follow the scientific method. Mongolic and Turkic languages (as well as some others, most notably the Tungusic ones) show a great deal of similar features that are thought to be areal features diffused a few thousand years ago (timeline off the top of my head). For example, Tuvan, a Turkic language, shows a great deal of Mongolian influence through borrowing and shared culture over a long period, which has resulted in Tuva and Western Mongolia being the two most active modern centers of Central Asian overtone singing.
In fact, many of these typological features found in Northern Asia are also present in Uralic, but even the Altai hypothesizers don't put Uralic in there. Uralic is the language family that "Finno-Ugric" is in, but we don't really use the latter term anymore because it isn't a primary family and there isn't good evidence that the Ugric and the Finno-Permic have much in common that separates them from the Samoyedic.
So to summarize: Mongolic languages have no proven genetic relationships to any non-Mongolic languages, but they share certain features with languages in a broad belt across Northern and Central Asia that are thought to be the result of areal contact.