Carney has joined Brighouse Town on a season-long deal.
The goalkeeper will continue to train with United during the week while gaining valuable first-team experience in the Northern Premier League. He spent time with Stocksbridge Park Steels, where his father Andy played, last term.
Haygarth embarks on his first loan move away from the club with Brentford's B team.
The 18-year-old, who can play in midfield and attack, has been with United since the age of nine.
The local lad moved into Neil Wood’s squad in the summer and came off the bench for the second half of the impressive 6-0 win against Salford City in the EFL Trophy.