Can one daven (pray) with dirt or urine on his clothing? I would like to know the exact laws because I have heard different peshutim on this. Please list sources.
R Eliezer Melamed writes here regarding dirt (early 3, end of 4, beginning of 5)
This should also be apparent in his dress; one’s clothes should be respectable, fitting for one who stands before the King.
If one is wearing disgraceful clothes, normally not worn on the street, such as dirty work clothes or shorts which he put on to work in his yard, it is better that he change his clothes, even if he will miss praying with a minyan. If he wears such clothing to pray, he will offend the respect of Heaven. Additionally, there is concern that he will not be able to concentrate on his prayer, since he will be thinking that everyone is staring at his disgraceful dress.
Those whose profession requires them to wear work clothes and it is difficult for them to change before praying are permitted to pray in their work clothes, because for them, these articles of clothing are not considered disgraceful. Nevertheless, in situations in which they have time to change their clothes, they should try to come to prayer in more respectable attire.
I read this as meaning that some dirt would be all right if you wouldn't mind standing in front of an important person with these clothes.
There is a prohibition to pray in front of tzo'a (from Berakhot 25a) but this refer primarily to human excrement. For mei raglayim (urine), the prohibition is only in front of a stream. R David Brofsky writes here
Mi-derabbanan, only an area wet enough that one who touches the area can dampen another material ("tofei'ach al menat le-hatpi'ach") is prohibited. The Arukh Ha-Shulchan argues that this prohibition is not specific to urine, and applies equally to other repulsive liquids [...]
For practical rulings and specifics CYLOR.