Is it true that the Companions used to consistently recite a specific supplication throughout Ramadan that would be answered by the arrival of the following Ramadan?

No specific supplication (du'a) has been authentically reported for Ramadan. What is prescribed for a Muslim is to strive in supplication generally, for Allah is Most Generous and Most Expansive. The Prophet (peace be upon him) promised that any supplication that is not a sin or cutting off family ties will be answered, either by being hastened, or by being stored for the Hereafter, or by warding off an equivalent evil. The most appropriate supplications to make in Ramadan are for salvation in the Hereafter, asking for Paradise, and seeking refuge from the Fire, as the Prophet (peace be upon him) said: "Around it we revolve." And among the best supplications for Laylat al-Qadr (the Night of Decree) is the saying: "O Allah, indeed You are an Pardoner, You love pardon, so pardon me."