In 1780, there was a Parliamentary inquiry into what went wrong in Burgoyne’s 1777 campaign. Below is a link to that document. It contains all sorts of interesting information from the British perspective. It really illuminated the limits of Burgoyne’s supply train and how his supply routes greatly influenced the route he took. In some ways, Burgoyne lost for two reasons. First he didn’t break through the American forces near Saratoga in September 1777. After that his supply lines were only thinly guarded, which allowed the American militia under General Lincoln to block his supply routes. That forced Burgoyne into one more attack in October, which also failed. Lacking provisions for his men, Burgoyne tried to retreat northwards, but that was a futile move as he lacked any British controlled supply depots in his immediate area. So this document (a primary source) will give you a good view about the supply problems (really lack of horses and carts) that hindered Burgoyne in his offensive.
And now the British side of Saratoga