Both sentences are correct. You can omit "his" if you want. Both sentences are equally good, so use whichever one you like better. There are sentences where "his" should not be omitted. For example, "He sat in the front row with Miss Smith, mouth half open." That could be confusing. Whose mouth is half open? His mouth or Miss Smith's mouth? In this case it's better to say, "He sat in the front row with Miss Smith, his mouth half open."

Annabelle is correct that the possessive pronoun 'his' can be omitted in both the example sentences however the second one "He sat in the front row, mouth wide open." is in my opinion incorrect if left the way it is because you have to further explain why his mouth is open. If you think about the image your sentence creates: a person sitting at the front of a class with his mouth wide open, you can see that it is quite strange, it makes one wonder why his mouth is open, so to be safe you should continue for example like this: He sat in the front row, mouth wide open in disbelief. or depending on the context of the sentence, for example in an everyday conversation: "So what did he do when he saw her?" "He sat in the front row, mouth wide open!!" it would be all right to use the sentence as you have written it because you can see that the speaker is emphasising the fact that his mouth was open. In my opinion using a sentence like that is only appropriate in informal spoken English, but never/rarely in written English.