It is raining, and they are trudging down the street.
In dictionary trudge defined as “To walk in a laborious, heavy-footed way”.
So when it’s raining, and the person walk quickly and laboriously, can I use the word “trudging”?
As some of the commenters noted, trudging implies moving along at a very slow, laborious pace, picking up one foot and placing it in front of the other, over and over and over again. It implies there is resistance or some sort of fatigue.
If you are wanting to say that the people walking in the rain were trying to move along quickly to get inside as soon as possible, you might use the word scurry, which means:
To hurry along; move hastily and precipitately; scamper.
It is raining, and they are scurrying down the street.
If you are wanting to say that the people walking in the rain were actually hindered by the rain or the wet street conditions, you could correctly use the word trudge, which means:
To walk in a laborious, heavy-footed way; plod.
It is raining, and they are trudging through the puddles and mud down the street.
Note, however, that the mental image that accompanies the last sentence is not one of people moving quickly.