When the Sorting Hat initially assigned Harry to House Slytherin, was it because of Harry's nature or was it actually detecting Voldemort's presence?
According to Pottermore, Harry was not a Hatstall
The only true Hatstalls known personally to Harry Potter were Minerva McGonagall and Peter Pettigrew.
Hatstall Defenition -
An archaic Hogwarts term for any new student whose Sorting takes longer than five minutes. This is an exceptionally long time for the Sorting Hat to deliberate, and occurs rarely, perhaps once every fifty years.
By definition despite the peice of Voldemorts soul and the debate Harry had with the Sorting Hat, he does not meet the requirements.
As to whether or not Voldemorts soul came into play in the deliberation, https://scifi.stackexchange.com/a/10988/29220 is the answer you want to take a look at.
He was a near Hatstall between Slytherin and Gryffindor but didn't take long enough to be a true Hatstall.
The Sorting Hat detected the presence of Voldemort's soul:
J.K. Rowling: "Also, I always imagine that the Sorting Hat detected the presence of that piece of [Voldemort's] soul when Harry first tried it on, because it's strongly tempted to put him in Slytherin.
From The Leaky Cauldon's interview.
However, it should be noted that it isn't wrong about Harry's Slytherin-esque personality traits as observed at multiple times. (He's determined, talented and has a penchant for rule-breaking).