או"ח ד:יז - רחיצת הפה בבוקר

There are those who are נוהג to wash their mouth in the morning because of the רירים (mucus or other such unpleasant debris) inside it, in order to have a clean mouth for davening, especially where one mentions the Name of Hashem. If the only water available, however, is not clean, and one is sensitive to that, then washing the mouth is not מעכב with regards to saying ברכות and the like.

Even one who normally washes their mouth every morning to have a clean mouth for davening still may not do so on a תענית ציבור.

References: שו"ע או"ח ד:יז, לבוש שם, כף החיים שם

It would seem that all of the above is not on the assumption that one anyway brushes their teeth with toothpaste every morning, in which case one would already be washing their mouth regardless. However, even one who does brush their teeth with toothpaste every morning, has, until reading this article, presumably only done so for hygiene. Hereon they can also invest one piece of their daily routine with halachic significance.