The home side equalised minutes before half time when Richards finished from a corner to make things level at the interval.
However, City regained their lead 8 minutes into the second half when Matt Bloomer headed home a corner.
15 minutes from time Bradshaw was brought down inside the box and the referee awarded a penalty. The 19-year-old picked himself up and scored with the kick to make it 3-1.
City finished the match with another goal in the closing stages. The goal was set up and scored by two juniors, with Jamie Heard crossing for Chris Hudson to head into the top corner.
The win pleased assistant manager Gary Barnett: "The lads have taken some stick after a couple of bad results and I said before the game that they needed to go out and prove a few people wrong.
"Fortunately they responded in the right manner and it was a good performance which means they deserve a bit of credit.
"Hopefully we can carry that forward when we travel to Goole this coming Saturday."
City line-up (4-3-3): Musselwhite; Petty, Mohan, Bloomer, Burton; Price (Heard), Philpott (Kerr), Mann; Bradshaw, Donaldson (Hudson), Peat (Holt).