Had? It still has.In the right circumstances, might not Turkey go for a little territorial readjustment at the expense of an Axis-allied neighboring country? Meaning Bulgaria, which still had a Turkish minority. It would have to be before Sept. 1944 when Bulgaria left the Axis.
OTL, Turkey had a non-aggression pact with the Nazis until 1944.
That's what was offered (or at least hinted) to Turkey in OTL, but they didn't consider it sufficient to enter the war. Incidentally, more Turks live in Northern Bulgaria than in the South and those who live there don't live on the border with Turkey (see map. The Gypsy population of Bulgaria is spread through out the country and Turkey is unlikely to want them.I was thinking that. Control of parts of southern Bulgaria with major Turkish and/or Roma populations (plus significant post-war influence in whatever Bulgarian polity emerges), plus perhaps the Aegean islands and some pieces of Thrace could put them on the Allied side somewhat earlier. If Churchill gets his way and Overlord is cancelled in favour of an invasion of the Balkans and southern France, the Turks might get pulled in with Allied bribes. A couple dozen divisions, even poorly armed and trained, would still cause a major headache for the Germans and could help secure liberated territory.
As for the effect of the Turkish invasion, while they add increased forces to the area (though not very well armed), their intervention will certainly lead to a significant resistance by Bulgaria (probably the only possible scenario where Bulgaria doesn't immediately surrender to an Allied invasion). The question is whether Bulgaria is able to resist the Turks with the help of Germany until they can surrender to the Soviets. If so, the situation will probably be close to OTL, as the Soviet Union would be unlikely to let a satellite state lose territory to a pro-Western country. Even if the Western allies capture Bulgaria with the help of Turkey, it might still be taken over the Communists, especially if the Turkish government is stupid enough to actually attempt to get "significant political influence" in Bulgaria.
As for the idea of giving Turkey parts of Greek Thrace (Thrace is a quite bigger region) and the Greek populated Dodekanese, backstabbing one's allies is not generally a good idea. It's quite possible that Greece will also be a Communist country in this time line.