depending on the dates then

most likely the Monmouth Workhouse
http://users.ox.ac.uk/~peter/workhouse/Monmouth/Monmouth.shtml

and some in the north of "West Dean" were no doubt served by the
"Ross Union" Workhouse ( or the Ruardean parish workhouse)....
http://users.ox.ac.uk/~peter/workhouse/Ross/Ross.html

and those over near Tintern / Chapel Hill were served by Chepstow
http://users.ox.ac.uk/~peter/workhouse/Chepstow/Chepstow.shtml