Is it necessary to respond to misgivings (waswas) about the impurity of gravel or sand adhering to the foot while going to the mosque, and is it sufficient to wash the foot with water to purify it if it is impure, or is it necessary to repeat the ablution (wudu)?

This act is a form of extremism (ghuluw), excessive scrupulosity (tanattu'), and succumbing to obsessive whispers (waswas). One must turn away from it. The default state of things is purity, and nothing becomes impure merely by doubt. Even street mud is considered pure, even if its impurity is strongly suspected, as long as impurity is not certain. And a small amount of impurity is excused if it is certain. Therefore, you must cast aside these obsessive whispers, and there is no need to wash the foot or perform ablution merely because of these doubts.