Is there an inyan to have a תענית דיבור for part of the day during shovavim?
Answer:
The Shovavim period is mentioned by halachic authorities, based on sources in Kaballah, as a virtuous time for penitence and specifically for fasting.
A taanit dibbur is a recommended practice. Although the idea is not mentioned in the Mishnah or Gemara, and even in the Rishonim (for whom fasting referred specifically to refraining from eating and drinking), it is noted by later authorities (see Mishnah Berurah 571:2; Shemiras Ha-Lashon Tevunah 2), and the Shovavim period is certainly a good time for somebody who wishes to strengthen himself in this matter to do so.
Best wishes.