I would mention Aiden's name first (before saying he or his). Otherwise, it isn't clear whether we are talking about one male character or two.
It also wasn't clear whether Aiden is trying to catch the woman before she goes through the portal or he is trying to catch up with others who are also after her, people he does not want to get hold of her--and she is already on the other side of the portal. So my insert (she?) is based on that confusion. If the others are on his side with a common goal, why must he be the one to catch her?
This pursuit is an interesting way to open overall, though.
Okay, here we go. Something new from me so have at it! :^D
Acrid smoke stung
his Aiden's eyes. His and Aiden's lungs burned as he sucked in another breath of tainted oxygen. As he raced Racing across the grassland at preternatural speed, he left small craters in the soft ground with every strike of his heels. His quarry managed to stay a few steps ahead of him and Aiden wasn't sure how much longer he could keep up this breakneck pace. Aiden knew he must catch the woman and the bundle in her arms before they (she?) reached the portal. Everything depended on it.