Is a man's prayer valid if his shoulders are uncovered, and does the ruling on uncovering the shoulders differ between prayer during Hajj and Umrah and other times?

There is no difference between prayer during Hajj and other prayers regarding the baring of the shoulder. Scholars differed concerning the ruling on praying in a single garment that does not cover the shoulders. The majority of scholars hold that the prayer is valid, though disliked, while Imam Ahmad and the Zahiris believe it is invalid. The evidence for the majority are hadiths indicating the permissibility of praying in a single garment without covering the shoulders, such as the hadith from Jabir ibn Abdullah from the Prophet, peace and blessings be upon him: "If it is wide, then wrap yourself in it; and if it is narrow, then use it as an izar (waist wrapper)." Imam An-Nawawi holds that the prohibition in the hadith of Abu Hurairah: "None of you should pray in a single garment, without anything on his shoulders," is for deprecation (tanzih) and not for prohibition (tahrim).