Green opened the scoring in the first minute, before goals from youth team players Will Atkinson and Nicky Featherstone either side of half time.
Barmby, who's been out since January with injury problems, got the fourth.
The home side pulled one back, before Barmby was substituted on the hour mark.
City XI line-up: Curtis Aspden; Mark Lynch, Matthew Plummer, Michael Byron, Tom Matthews; James Bennett, Stuart Green, Russell Fry, Will Atkinson; Nicky Featherstone, Nick Barmby.